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AN   INTRODUCTION 
t 

1  TO   THE 

•      INFINITESIMAL   CALCULUS 


AN    INTRODUCTION 


TO   THE 


INFINITESIMAL   CALCULUS 

NOTES  EOR  THE   USE   OE   SCIENCE 
AND    ENGINEEKING    STUDENTS 


BY 

H.  S.  CAKSLAW,  Sc.D. 

PROFESSOR  OF  MATHEMATICS  IN  THE   UNIVERSITY  OF  SYDNEY 

FORMERLY   FELLOW   OF   EMMANUEL   COLLEGE,   CAMBRIDGE 

AND   LECTURER   IN   MATHEMATICS   AT   THE   UNIVERSITY   OF   GLASGOW 


SECOND    EDITION 


LONGMANS,   GEEEN   AND    CO. 

39  PATERNOSTER  ROW,   LONDON 
NEW   YORK,   BOMBAY  AND  CALCUTTA 

'  1912 


:-,.  ^r^ 


CSS 

Wis. 


PEEFACE  TO  THE  FIEST  EDITION 

These  introductory  chapters  in  the  Infinitesimal  Calculus  were 
lithographed  and  issued  to  the  students  of  the  First  Year  in 
Science  and  Engineering  of  the  University  of  Sydney  at  the 
beginning  of  last  session.  They  form  an  outline  of,  and  were 
meant  to  be  used  in  conjunction  with,  the  course  on  The  Elements 
of  Analytical  Geometry  and  the  Infinitesimal  Calculus,  which  leads 
up  to  a  term's  work  on  Elementary  Dynamics. 

The  standard  text-books  amply  suffice  for  the  detailed  study 
of  this  subject  in  the  second  year,  but  the  absence  of  any  dis- 
cussion of  the  elements  and  first  principles  suitable  for  the  first 
year  work,  was  found  to  be  a  serious  hindrance  to  the  work  of 
the  class.  For  such  students  a  separate  course  on  Analytical 
Geometry,  without  the  aid  of  the  Calculus,  is  not  necessary,  and 
the  exclusion  of  the  methods  of  the  Calculus  from  the  analytical 
studj^  of  the  Conic  Sections  is  quite  opposed  to  the  present 
unanimous  opinion  on  the  education  of  the  engineer.  It  has 
been  our  object  to  present  the  fundamental  ideas  of  the  Calculus 
in  a  simple  manner  and  to  illustrate  them  by  practical  examples, 
and  thus  to  enable  these  students  to  use  its  methods  intelli- 
gently and  readily  in  their  Geometrical,  Dynamical,  and  Physical 
work  early  in  their  University  course.  This  little  book  is  not 
meant  to  take  the  place  of  the  standard  treatises  on  the  subject, 
and,  for  that  reason,  no  attempt  is  made  to  do  more  than  give 
the  lines  of  the  proof  of  some  of  the  later  theorems.  As  an 
introduction  to  these  works,  and  as  a  special  text-book  for  such 
a  "  short  course  "  as  is  found  necessary  in  the  engineering  schools 
of  the  Universities  and  in  the  Technical  Colleges,  it  is  hoped  that 
it  may  be  of  some  value. 


292611 


vi  PEEFACE 

In  the  preparation  of  these  pages  I  have  examined  most  of 
the  standard  treatises  on  the  subject.  To  Nernst  and  Schonflies' 
Lehrbuch  der  Differential-  und  Integral-Eechmmg,  to  Vivanti's 
Complementi  di  Matematica  ad  uso  dei  Chemici  e  dei  Naturalistic  to 
Lamb's  Infinitesimal  Calcidus,  and  to  Gibson's  Elementary  Treatise 
on  the  Calculus,  I  am  conscious  of  deep  obligations.  I  should 
also  add  that  from  the  two  last-named  books,  and  from  those 
of  Lodge,  Mellor,  and  Murray,  many  of  the  examples  have  been 
obtained. 

In  conclusion,  I  desire  to  tender  my  thanks  to  my  Colleagues 
in  the  University  of  Sydney,  Mr.  A.  Newham  and  Mr.  E.  M. 
Moors,  for  assistance  in  reading  the  proof-sheets ;  to  my  students, 
Mr.  D.  R.  Barry  and  Mr.  K.  J.  Lyons,  for  the  verification  of 
the  examples;  also  to  my  old  teacher.  Professor  Jack  of  the 
University  of  Glasgow,  and  to  Mr.  D.  K.  Picken  and  Mr.  R.  J. 
T.  Bell  of  the  Mathematical  Department  of  that  L^niversity,  by 
whom  the  final  proofs  have  been  revised. 

H.  S.  CARSLAW. 

The  University  of  Sydney, 
J%me,  1905. 


PREFACE  TO  THE  SECOND  EDITION 

The  principal  change  in  this  edition  will  be  found  in  the 
treatment  of  the  exponential  and  logarithm.  Six  years  ago 
few  students  began  the  study  of  the  Calculus  without  having 
already  completed  a  course  in  Algebra,  including  the  Theory 
of  Infinite  Series.  It  is  now  realised  that  in  making  this 
demand  the  mathematical  teacher  was  asking  more  than  was 
necessary.  The  principles  underlying  the  Calculus,  in  so  far 
as  they  can  be  examined  in  such  a  course  as  this,  offer  little 
difficulty.  No  more  than  an  elementary  knowledge  of  Algebra 
and  Trigonometry  is  required  for  their  discussion ;  and  a  real 
grasp  of  the  meaning  of  differentiation  and  integration  can  be 
obtained  by  very  many  to  whom  the  subject  of  Infinite  Series 
would  appear  extremely  obscure. 

These  altered  conditions  have  allowed  me  to  place  the  older 
proofs  of  the  theorems  regarding  the  differentiation  of  e*  and  log  x 
in  an  Appendix,  and  I  have  introduced  into  the  text  one  of  the 
simpler  methods,  in  which  use  is  made  of  the  Logarithm  Tables. 
In  this  discussion  I  have  followed  the  lines  laid  down  by  Love 
in  his  Elements  of  the  Differential  ami  Integral  Calculus.  However 
it  seemed  worth  while  to  carry  the  numerical  work  a  little 
further,  with  the  help  of  8-Figure  and  15-Figure  Tables. 
The  student  is  apt  to  imagine  that  4-Figure  and  even  7-Figure 
Tables  give  a  more  accurate  result  than  they  frequently  afford. 

The  other  changes  that  need  be  mentioned  are  the  addition 
of  a  section  on  Repeated  Differentiation,  and  one  on  Fluid 
Pressure.  A  number  of  easy  examples  and  of  graphical  illus- 
trations have  also  been  inserted. 

H.  S.  CARSLAW. 

Sydney,  December,  1911. 


CONTENTS 


CHAPTER  I 
THE  ANALYTICAL  GEOMETRY  OF  THE  STRAIGHT  LINE 

SECT.  PAOE 

1.  Cartesian  Co-ordinates         .-.----  i 

2.  The  Co-ordinates  of  the  Point  at  which  a  Line  is  divided  in 

a  given  Ratio -         -  2 

3.  The  Equation  of  the  First  Degree        .         .         -         .         .  3 

4.  Drawing  Straight  Lines  from  their  Equations     -         -         -  6 

5.  The  Gradient  of  a  Line 7 

6.  Different  Forms  of  the  Equation  of  the  Straight  Line          -  8 

7.  The  Perpendicular  Form •  -  9 

8.  The  Point  of  Intersection  of  two  Straight  Lines  -         -         -  9 

9.  The  Angle  between  two  Straight  Lines        -         -         -         -  10 
10.  The  Length  of  the  Perpendicular  from  a  given  Point  upon  a 

Straight  Line 13 

Examples  on  Chapter  I. 15 


CHAPTER  II 
THE  MEANING  OF  DIFFERENTIATION 

11.  The  Idea  of  a  Function        ...---.  17 

12.  Illustrations  from  Physics  and  Dynamics    -         -         -         -  17 

13.  The  Fundamental  Problem  of  the  Differential  Calculus        -  19 

14.  Rectilinear  Motion       -         -         - 19 

15.  Limits.     Differential  Coefficient 21 

16.  Geometrical  Illustration  of  the  Meaning  of  the  Differential 

Coefficient 23 


X  CONTENTS 

SKCT.  PAGE 

17.  Approximate  Graphical  Determination  of  the  Differential 

Coefficient        .         -         .         - 25 

18.  Repeated  Differentiation      -         -         -         -         -         -         -  25 

Examples  on  Chapter  II. 29 


CHAPTER  III 

DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS;    AND  SOME 
GENERAL  THEOREMS  ON  DIFFERENTIATION 

19.  Differentiation  of  a)'\  when  n  is  a  Positive  Integer       -         -  31 " 

20.  Some  General  Theorems — 

I.  Differentiation  of  a  Constant  -----  33 
II,  Differentiation  of  the  Product  of  a  Constant  and  a 

Function -  33 

IIL  Differentiation  of  a  Sum  ------  34 

IV.  Differentiation  of  the  Product  of  Two  Functions     -  34 

V.  Differentiation  of  the  Quotient  of  Two  Functions    -  35 

VI.  Differentiation  of  a  Function  of  a  Function     -         -  36 

21.  Differentiation  of  .v'\  when  7i  is  any  Positive  or  Negative 

Number-         ---------  39 

Examples  on  Chapter  III.         ------  43 


CHAPTER  IV 
THE  DIFFERENTIATION  OF  THE  TRIGONOMETRIC  FUNCTIONS 

22.  Differentiation  of  sin  x 45 

23.  Differentiation  of  cos  .t^         -         -         -         -         -         -         -  46 

24.  Differentiation  of  tan  x         -------  47 

25.  Geometrical  Proofs  of  these  Theorems          -        -         -         -  47 

26.  The  Graphs  of  the  Trigonometrical  Functions     -         -         -  48 

27.  Differentiation  of  sin-^«?      ----...  51 

28.  Differentiation  of  cos"^^ -  54 

29.  Differentiation  of  tan""^^ 57 

Examples  on  Chapter  IV.        ------  59 


CONTENTS  xi 


CHAPTER  V 


THE    EXPONP^NTIAL    AND    LOGARITHMIC    FUNCTIONS; 
MAXIMA  AND  MINIMA  ;   PARTIAL  DIFFERENTIATION 


SECT. 


PAGE 


30.  Introductory        - 61 

31.  The  Expressions  n+-j    and  ?ilogio(  1+-)         -         -         -  63 

32.  Differentiation  of  logjo-r       -------  65 

33.  Differentiation  of  log  jp  and  log„.i'         -----  66 

34.  Differentiation  of  e* -         -         -  68 

35.  Differentiation  of  a*^     -         - 69 

36.  Logarithmic  Differentiation  - 69 

37.  Differentiation  of  e"""^  sinbx        -         -         -         -         -         -  70 

38.  Maxima  and  Minima  of  Functions  of  one  Variable      -         -  71 

39.  Points  of  Inflection      --------  73 

40.  Partial  Differentiation  -__----  74 

41.  Total  Differentiation    -         -         _         -         -         -         -         -  75 

42.  Differentials         ---------  76 

Examples  on  Chapter  V. -  78 

CHAPTER  VI 
THE  CONIC  SECTIONS 

43.  Introductory 81 

44.  Discussion  of  the  Parabola  and  Examples    -         -         -         -  81 

45.  Discussion  of  the  Ellipse  and  Examples       -         -         -         -  84 

46.  Discussion  of  the  Hyperbola  and  Examples  -         -         -  88 

CHAPTER  VH 
THE  INTEGRAL  CALCULUS— INTEGRATION 

47.  Introductory -         -         -  92 

48.  Standard  Integrals 94 

49.  Two  General  Theorems 96 


xii  CONTENTS 

SECT.  PAGE 

50.  Integration  by  Substitution 98 

51.  Integration  by  Substitution  {continued)       .         -         .         .  100 
62.    Integration  by  Parts 102 

Examples  on  Chapter  VII. 104 


CHAPTER   VIII 
THE  DEFINITE  INTEGRAL  AND  ITS  APPLICATIONS 

53.  Introductory 107 

54.  Areas  of  Curves.     The  Definite  Integral  as  an  Expression 

for  the  Area 107 

55.  The  Definite  Integral  as  the  Limit  of  a  Sum        -         -         -  111 

56.  The  Evaluation  of  a  Definite  Integral  from  its  Definition 

as  the  Limit  of  a  Sum  - 113 

57.  Properties  of  r' f{x)dx 114 

J  Xq 

58.  Application  of  the  Definite  Integral  to  Fluid  Pressure         -       116 

59.  Application   of  the   Definite   Integral   to  Areas  in  Polar 

Co-ordinates -         -  119 

60.  Application  of  the  Definite  Integral  to  Lengths  of  Curves  -  120 

61.  Application  of  the  Definite  Integral  to  Volumes  of  Solids    -  121 

62.  Application  of  the  Definite  Integral  to  Surfaces  of  Solids  of 

Revolution       -         - 122 

63.  Application  of  the  Definite  Integral  to  the  Centre  of  Gravity 

of  a  Solid  Body 123 

64.  Application   of  the   Definite   Integral  to  the  Moment  of 

Inertia  of  a  Solid  Body  -         -         -         -         -         -       125 

Examples  on  Chapter  VIII. 126 

Appendix 129 

Answers 131 


BLANK  PAGE  FOR  IMPORTANT  RESULTS 
IN  DIFFERENTIATION 


BLANK  PAGE  FOR  IMPORTANT  RESULTS 
IN  INTEGRATION 


CHAPTER  I 

THE   ANALYTICAL   GEOMETRY  OF   THE   STRAIGHT   LINE 

§  1.   Cartesian  Co-ordinates. 

The  position  of  a  point  on  a  plane  may  be  fixed  in  different 
ways.  In  particular  it  is  determined  if  its  distances  from  two 
fixed  perpendicular  lines  in  the  plane  are  known,  the  usual  con- 


P  {^,y) 


Fig.  1. 

ventions  with  regard  to  sign  being  adopted.  These  two  lines 
Qx  and  Oy  are  called  the  axes  of  x  and  y  \  and  the  lengths  OM 
and  ON,  which  the  perpendiculars  from  the  point  P  cut  off  from 
the  axes,  are  called  the  co-ordinates  of  the  point  P  and  denoted 
C.C.  A 


2.  TFE  AI^>ALYTICAL  GEOMETEY 

by  X  and  y.  OM  and  ON  are  taken  positive  or  negative  accord- 
ing as  they  are  measured  along  Ox  and  O?/,  or  in  the  opposite 
directions.  OM  is  called  the  "  abscissa  "  of  P  and  MP  is  called 
the  ''m-dinate"  of  P. 

Ex.  1.    Mark  on  a  piece  of  squared  paper  the  position  of  the  points 
(±2,  ±3).. 

2.  Prove  that  the  distance  between  the  points  (2,  3),  and  (-2,  -3)  is 
2\/l3. 

3.  Prove  that  the  distance  d  between  the  points  (x^,  y-j),  (x^,  ya)  ia 
^ven  by  d2  =  (xi  -  X2)2  +  {y,  -  y.^\ 

4.  Prove  that  the  co-ordinates  of  any  point  (x,  y)  upon  the  circle  whose 
centre  is  at  the  point  (a,  b)  and  whose  radius  is  c  satisfy  the  equation 

(x-a)2  +  (y-b)2=c2. 

§  2.   The  Co-ordinates  of  a  Point  dividing  the  Line  joining  two 
given  Points  in  a  given  Ratio  1 :  m. 


y 

P2 

P      ^^ 

p. 

^^^^ 

L 

1 

K 

H 

0 

K 

/I,           M         Mj 

• 

k' 

► 

y' 

Fig.  2. 


Let  Pj  and  P2  be  the  two  given  points  {x^,  y^),  {x^,  y^;  and 
let  V{x,  y)  divide  P^Pg  in  the  ratio  I :  m  (see  Fig.  2). 
Draw  PjMj,  PM  and  P2M2  perpendicular  to  Ox. 


OF  THE  STRAIGHT  LINE 

Also  draw  P^HK  and  PL  parallel  to  Ox. 

Let  these  lines  meet  PM  and  P2M2  in  H,  K  and  L. 

Since 
we  have 


P,H 
PL 

pp. 

X- 

-X    m 

x{l  +  m)  =  lx.2 

+  mxy 

1x9  +  mxi 

.'.   x  = 


1  +  m 


Similarly  y-^^itm^'" 

These  are  the  co-ordinates  of  the  internal  point  of  section. 
Those  of  the  external  point  may  be  found  in  the  same  way  to  be 

Ixn  -  mx. 

a;=— 4 *- 

/  -m 

and  yjy^^^zm. 

^        l-m 

As  a  particular  case  of  this  theorem  the  co-ordinates  of  the 
middle  point  of  the  line  joining  (Xj,  y^  and  (xg,  72)  are 

^1+^2  and  yi-^y^. 

2  2 

Ex.  1.  Prove  that  the  co-ordinates  of  the  middle  point  of  the  line  which 
cuts  ofif  unit  length  from  Oa;  and  Oy  are  J  and  |. 

2.  Find  the  co-ordinates  of  the  points  of  trisection  of  this  hne,  and  also 
of  the  points  which  divide  it  externally  in  the  ratio  1  :  2. 

3.  Prove  that  the  C.G.  of  the  triangle  whose  angular  points  are  (2,  1), 
(4,  3),  (2,  5)  is  the  point  (  s'  ^ )  '  ^'^d  give  the  general  theorem. 

§  3.  The  Equation  of  the  First  Degree  represents  a  Straight 
Line. 

If  the  point  P  moves  along  a  curve,  the  co-ordinates  of  the 
point  are  not  independent  of  each  other.  In  mathematical 
language  "y  is  a  function  of  re";  and  we  speak  of  y=-f{x)  as  the 
equation  of  the  curve,  meaning  that  all  the  points  whose  co- 
ordinates satisfy  this  equation  lie  upon  the  curve,  and  that  the 
co-ordinates  of  all  points  upon  the  curve  satisfy  the  equation. 


4  THE  ANALYTICAL  GEOMETRY 

For  example,  the  equation  of  the  circle  with  its  centre  at  the 
point  {a,  h)  and  radius  c  is  {x-  af  +  {y  -  hy  =  c^.     (Cf.  §  1,  Ex.  4.) 

The  same  ideas  are  employed  in  Solid  Geometry  :  the  surface 
of  a  solid  is  represented  by  an  equation  satisfied  by  the  co- 
ordinates of  the  points  lying  upon  it;  and  straight  lines  and 
curves  are  given  by  simultaneous  equations.  The  geometrical 
properties  of  curves  and  surfaces  may  often  be  obtained  by 
discussing  their  equations.  This  branch  of  mathematics  is  called 
Analytical  Geometry. 

The  simplest  equation  in  the  two  variables  ic,  y  is  that  of  the 
first  degree  ax  +  hy  +  c  =  Qf, 

a,  h  and  c  being  constants. 

For  example,  take  the  equation 
a; +  2^  =  4. 

By  assigning  any  value  to  x  and  solving  the 
equation  for  y,  we  obtain,  as  in  the  accompany- 
ing table,  the  co-ordinates  of  any  number  of 
points  upon  the  locus.  Plotting  these  points 
upon  squared  paper  in  the  usual  way,  we  see 
that  they  all  lie  upon  a  straight  line  ;  and,  so 
far  as  our  measurements  could  be  relied  upon, 
we  could  verify  that  the  co-ordinates  of  any 
point  upon  this  line  would  satisfy  the  equation. 

We  proceed  to  prove  that  this  is  true  in  general :  in  other 
words,  that  all  the  points  whose  co-ordinates  satisfy  the  equation 

ax  +  hy-\-c  =  0 
lie  upon  one  and  the  same  straight  line,  and  that  the  co-ordinates  of  all 
points  upon  this  straight  line  satisfy  the  equation. 

(i.)  We  consider  first  of  all  the  equation 

y  =  mx,  (1) 

m  being  any  real  number. 

Let  P  be  any  point  whose  co-ordinates  x,  y  satisfy  this  equation. 
(Cf.  Fig.  1.) 

Draw  PM  perpendicular  to  the  axis  of  x  and  join  OP. 

Let  OP  make  an  angle  6  with  the  positive  direction  of  the 

axis  of  X. 

.      z)     MP     y 
Then  tan  6  =  ^, ,  =  -  =  m. 

OM     X 


X 

y 

-3 

3-5 

-  2 

3 

-1 

2-5 

0 

2 

1 

1-5 

2 

1 

3 

•5 

OF  THE  STEAIGHT  LINE  5 

Therefore  the  point  P  lies  upon  the  straight  line  through  the 
origin  which  makes  an  angle  whose  tangent  is  m  with  the 
positive  direction  of  the  axis  of  x. 

As  the  number  m  is  a  given  number,  this  line  is  a  definite 
straight  line. 

Now  let  us  take  any  point  upon  this  straight  line  and  draw 
the  perpendicular  from  that  point  to  the  axis  of  x. 

It  will  be  seen  that  the  co-ordinates  of  the  point  satisfy  the 
given  equation. 

It  follows  that  every  point  whose  co-ordinates  satisfy  the  equation 

y  =  mx 

lies  upon  a  certain  straight  line  through  the  origin,  and   that  the 
co-ordinates  of  every  point  upon  this  line  satisfy  the  equation. 

If  m  >  0,  the  line  will  be  in  the  first  and  third  quadrants. 

If  m=  0,  the  line  is  the  axis  of  x,  and  if  m=  oo,  the  line  is  the  axis  of  y. 

If  m  <  0,  the  line  is  in  the  second  and  fourth  quadrants. 

(ii.)  We  next  consider  the  equation 

y  =  mx  +  71,  (2) 

where  m  and  n  are  any  real  numbers. 

For  any  value  of  x  there  is  one  and  only  one  value  of  y.  This 
value  is  greater  by  n  than  that  for  the  corresponding  point  on 
the  straight  line  given  hy  y  =  mx. 

Hence,  to  obtain  all  the  points  whose  co-ordinates  satisfy  equation  (2), 
we  have  only  to  lengthen  the  m'dinates  of  all  the  points  on  the  straight 

^*^^  y  =  mx 

hy  an  amount  n. 

In  other  words,  we  have  only  to  move  this  whole  line  parallel 
to  itself  through  the  distance  n  in  the  direction  of  the  axis 
of  y. 

Or,  more  simply,  we  have  to  draw  the  parallel  through  the 
point  (0,  n)  to  the  line  y  =  r)ix. 

lin>0,  the  point  lies  on  the  positive  portion  of  the  axis  of  y ; 
if  ?i<  0,  it  lies  on  the  negative  portion. 

The  co-ordinates  of  all  points  upon  this  line  satisfy  equation  (2) ; 
and  the  co-ordinates  of  all  points  which  do  not  lie  upon  this  line 
do  not  satisfy  equation  (2). 


6  THE  ANALYTICAL  GEOMETRY 

(iii.)  Finally  we  consider  the  equation 

ax  +  by  +  c  =  0.  (3) 

If  b  =  0,  X  remains  constant  and  the  equation  represents  a  line 
parallel  to  the  axis  of  y. 

If  b  =1=  0,  we  can  write  the  equation  in  the  form 


(-|)-(-|> 


On  putting  m=  -  t    and     n^  -ji 

this  becomes  y  =  mx  +  n, 

the  form  we  have  discussed  in  (ii,). 

It  follows  that  the  equation 

ax  +  by  +  c  =  0 
always  represents  a  straight  line. 

For  this  reason  the  equation  of  the  first  degree  is  usually 
called  a  linear  equation. 

In  the  above  discussion  we  started  with  the  equation,  and  found  that 
the  locus  which  it  represents  is  a  straight  line. 

If  a  straight  line  is  given,  we  can  easily  show  that  the  co-ordinates  of 
any  point  upon  it  satisfy  a  linear  equation,  which  can  always  be  obtained. 

If  the  line  is  parallel  to  the  axis  of  x,  it  is  clear  that  the  ordinates  of  all 
points  upon  it  are  the  same.     Its  equation  is  thus  y  =  const. 

If  it  is  parallel  to  the  axis  of  y,  its  equation  is  a;  =  const. 

If  it  makes  an  angle  6  with  the  positive  direction  of  the  axis  of  x,  and 
cuts  off  an  intercept  w  from  the  axis  of  y,  we  have  seen  that  its  equation  is 
y  =  mx  +  n,  where  tan  6  =  m. 

We  thus  speak  of  the  equation  of  the  given  straight  line,  and  we  know 
that  it  is  always  of  the  form 

ax  +  by  +  c  =  0, 
where  a,  b  and  c  are  constants  which  arise  in  specifying  the  line. 

§4.   Drawing  Straight  Lines  from  their  Equations. 

In  the  last  article  we  have  shown  that  the  equation  of  the 
first  degree  represents  a  straight  line.  In  plotting  the  locus 
given  by  such  an  equation,  we  do  not  now  need  to  obtain  a  table 
of  values  of  x  and  y,  as  we  did  above  in  the  example  x  +  2y  =  4:. 
Two  points  fix  a  straight  line.  Therefore  we  have  only  to  find 
two  points  whose  co-ordinates  satisfy  the  equation.  The  most 
convenient  points  are  those  where  the  line  cuts  the  axes,  and 
these  are  found  by  putting  x  =  0  and  y  =  0,  respectively,  in  the 
equation. 


OF  THE   STRAIGHT  LINE 

Ex.1.    Draw  the  lines      (i.)  a;  =  0,     x  =  l,     x=-l; 

(ii.)  y  =  0,     y=2,     y=-2; 

{iii.)  x  +  y  =  0,     x  +  y  =  l; 

(iv.)  y  =  2x,        y—2x  +  3; 


(v.,  M=i, 


x    y 


=  h 


2.  Determine  whether  the  point  (2,  3)  is  on  the  line 

4x  +  3y=l5. 

3.  What  is  the  condition  that  the  point  (a,  h)  should  lie  upon  the  line 

ax  +  hy  =  2ah'! 

§  5.   The  Gradient  of  a  Line. 

When  we  speak  of  "the  gradient"  of  a  road  being  1  in  200 
we  usually  mean  that  the  ascent  is  1  foot  vertical  for  200  feet 
horizontal.  This  might  also  be  called  the  slope  of  the  road. 
The  same  expression  is  used  with  regard  to  the  straight  line. 
The  "gradient"  or  the  "slope"  of  a  straight  line  is  its  rise  per 
unit  horizontal  distance; 
or  the  ratio  of  the  increase 
in  y  to  the  increase  in  x 
SiS  we  move  along  the  line. 
This  is  evidently  the  same 
at  all  points  of  the  straight 
line,  and  is  equal  to  the 
tangent  of  the  angle  the 
line  makes  with  the  axis 
of  X  measured  in  the  posi- 
tive direction. 

To  save  ambiguity  it  is 
well  to  fix  upon  the  angle 
to  be  chosen,  and  in  these 

pages  it  will  be  convenient  to  consider  the  line  as  always  drawn 
upward  in  the  direction  ^  >  0  (Fig.  3),  and  thus  to  restrict  the 
angle  (f>  to  lie  between  0°  and  180°.  It  is  convenient  to  speak 
of  the  line  as  drawn  in  the  positive  direction  in  such  a  case. 

When  0  <  (^  <  ^  the  gradient  is  positive. 
When  ^  <  </)  <  TT  the  gradient  is  negative. 
Ex.    Write  down  the  values  of  <p  for  the  lines  in  §  4  (i.). 


Fig.  3. 


k 


8  THE  ANALYTICAL  GEOMETRY 

§  6.   Different  Forms  of  the  Eciuation  of  the  Straight  Line. 

In  the  preceding  articles  we  have  shown  that  the  equation 
ax  +  by  +  c  =  0 
represents  a  straight  line,  and  we  have  seen  how  the  line  may 
be  drawn  when  its  equation  is  given.     We  shall  now  show  how 
to  obtain  the  equation  of  the  line  when  two  points  upon  it  are  given. 

Let  (x-^,  y-^),  (x^,  y^  be  the  two  given  points.  Let  {x,  y)  be 
the  co-ordinates  of  any  point  upon  the  line.  Then  it  is  clear 
(cf.  Fig.  2)  that 

tzi  is  equal  to  the  gradient  of  the  line, 

and  that      ^^"^^  is  also  equal  to  the  gradient  of  the  line. 
Thus  we  have  the  equation 

tV  i/^i  tAj£)  (//-I 

between  the  co-ordinates  {x,  y)  of  the  representative  point  and 
the  co-ordinates  {x^,  3/i)(^2»  V^)  ^^  ^^^  fixed  points.  This  is  the 
equation  of  the  straight  line  through  these  points.  It  is  more 
conveniently  written 

x-x^^y-y^  ^. 

^1-^2  y^-y2 

It  follows  from  the  above  argument,  or  can  be  proved  inde- 
pendently, that 

The  equation  of  the  line  through  {x^^  y^),  making  an  angle  </>  with 
the  axis  of  x,  is 

|^  =  tan.^;  (B> 

and  that 

The  equation  of  the  line  which  cuts  off  a  length  c  from  the  axis  of  y, 
and  is  inclined  at  an  angle  whose  tangent  is  m  to  the  axis  of  x,  is 

y  =  mx  +  c;  (C) 

and  that 

The  equation  of  the  line  which  cuts  off  intercepts  a  and  h  from  the 
axis  of  X  and  y  is 

-  +  |=1.  (D) 

ah  .       ^ 


OF  THE  STRAIGHT  LINE 


9 


Ex.  1.  Write  down  the  equations  of  the  lines  through  the  following  paira 
of  points:  (1,  1),  (1,  -1);  (1,  2),  (-1,  -2);  (3,  4),  (5,  6) ;  {a,  b),  {a,  -h). 

2.  Find  the  equations  of  the  lines  through  the  point  (3,  4)  with  gradient 
±  5,  and  draw  the  lines. 

3.  The  lines  y=:x  and  y^lx  form  two  adjacent  sides  of  a  parallelogram, 
the  opposite  angular  point  being  (4,  5).  Find  the  equations  of  the  other 
two  sides  ;  and  of  the  diagonals. 

4.  Write  down  the  equations  of  the  lines  making  angles  30°,  45°,  60°, 
120°,  135°,  and  150°  with  the  axis  of  x,  which  cut  this  axis  at  unit  distance 
from  the  origin  in  the  negative  direction. 

§7.  The  "Perpendicular"  Form  of  the  Equation  of  the 
Straight  Line. 

A  straight  line  is  determined  when  the  length  of  the  perpen- 
dicular upon  it  from  the 
origin,   and  the  direction 
of  this  perpendicular  are 
given. 

Let  ON  be  the  perpen- 
dicular, p,  upon  the  line 

Let  the  angle  between 
ON  and  Ox  be  a,  this 
angle  lying  between  0 
and  27r  (cf.  Fig.  4). 

Then  N  is  the  point 
{'p  cos  a,  jp  sin  a). 


Fig.  4. 


y  -p  sm  a 
x-p  cos 

This  reduces  to 


Using  the  form  (B)  of  §  6,  the  equation  of  the  line  becomes 

a      ,         ,      ,        /        7r\  COS  a 

-  =  tan  <h  =  tan    a  +  ?:    =  — -. — . 

a  V         2/  sma 

X  COS  a  +  ?/  sin  a  =p.  (E) 

N.B.—ThQ  quantity  p  is  to  be  taken  always  positive,  and  the 
angle  a  is  the  angle  between  Ox  and  ON. 

§  8.   The  Point  of  Intersection  of  Two  Straight  Lines. 
Since  the  point  of  intersection  of  the  two  lines 

ax  +  by  +c  =0, 

a'x  +  b'y  +  c=-0 

lies  on  bpth  lines,  its  co-ordinates  x,  y  satisfy  both  equations. 


10  THE  ANALYTICAL   GEOMETRY 

Solving  the  equations  we  have 

X  y  1 


he'  -  b'c     ca'  -  c'a     ah'  -  a'h 

It  is  clear  that  if  ah'  -  a'h  =  0, 

and   neither   of   the   other   two   denominators   vanish,    the   co- 
ordinates X,  y  are  infinite,  and  the  lines  are  parallel. 

If  in  addition  ca'  -  c'a  =  0, 

1  a     h     c 

we  have  _  =- =- 

a     h     c 

and  the  third  denominator  he'  -  h'c  also  vanishes. 

In  this  case  the  two  equations  are  not  independent,  and  they 
really  represent  the  same  straight  line. 

Ex.  1.    Find  the  co-ordinates  of  the  point  of  intersection  of  the  lines 
2x+   y  =  4, 
x  +  2y  =  6. 
Illustrate  your  result  by  a  diagram. 

2.  Find  the  equations  of  the  lines  through  (2,  3)  parallel  to 

3x±4y  =  5. 

3.  Find  the  co-ordinates  of  the  angular  points  of  the  triangle  whose 
sides  are  given  by  x+  y  =  2,  (1) 

3x-2y=l,  (2) 

4x  +  3y  =  24.  (3) 

Also  find  the  equations  of  the  medians  of  this  triangle  and  the  co- 
ordinates of  its  C.G. 

§  9.   The  Angle  between  Two  Straight  Lines  whose  Ecluations 
are  given. 

When  one  of  the  lines  is  parallel  to  the  axis  of  y,  the  angle 
between  them  can  be  readily  found. 

In  all  the  other  cases  the  equations  can  be  reduced  to  the  forms 

y  =  mx  +c,  (1) 

y^m'x  +  c'.  (2) 

Also  the  angle  between  these  lines  is  the  same  as  the  angle 
between  the  lines  ^^^^  ^3^ 

and  y  =  m'x,  (4) 

which  pass  through  the  origin  and  are  parallel  to  the  given  lines. 


OF  THE  STEAIGHT  LINE 


11 


Let  OQ,  OQ'  (cf.  Fig.  5)  be  the  positive  directions  of  the 
lines  (3)  and  (4),  with  gradients  m  and  m',  respectively. 


Let 

LxOQ=cf>      (0<<^<7r) 

and 

LxOQ;  =  ct>'.      (0<<^'<7r) 

Then 

tan<^  =m 

and 

tan  <f>'  =  m'. 

Fig.  5. 

Let  cf)  be  the  larger  of  these  two  angles,  and  let  6  be  the  angle 
between  the  lines. 

Then  e  =  ^-<j>'. 

Therefore  tan  6  =  tan  (<^  -</>') 

tan  (f>  -  tan  4>' 


1  +  tan  </>  tan  <^' 
m-m' 


1  +  mm' 
Since  there  is  only  one  angle  0  less  than  two  right  angles, 
which  satisfies  this  equation,  it  can  be  solved  without  ambiguity. 


If ,  is  positive,  the  anele  6  is  acute. 

1+mm      ^  * 

If 7  is  negative,  it  is  obtuse. 

1  +  mm  ° 


12        THE  ANALYTICAL  GEOMETEY 

fVe  ham  thus  proved  that  the  absolute  value  of 

m-m' 
1  +  mm' 
is  equal  to  the  tangent  of  the  acute  angle  hetioeen  the  lines 

y^mx  +c, 
y  =  m'x  +  c'. 

In  practice  it  is  unnecessary  either  to  draw  the  lines,  or  to  consider 
which  has  the  greater  slope.  Taking  the  lines  in  any  order,  we  need  only 
calculate  the  absolute  value  of  the  expression 

m-m' 
1  +  mm' ' 
The  acute  angle  between  the  lines  can  then  be  written  down. 

It  follows  that 

(i.)  The  lines  are  parallel,  if  m  =  m' ; 

(ii.)  The  lines  are  perpendicular,  if  mm'  +  1=0. 

When  the  equations  are 

ax  +by  +c  =0, 

and  .  a'x  +  b'y  +  c'  =  0, 

ah'  -  a'h 


aa'  +  hh' 
is  equal  to  the  tangent  of  the  acute  angle  between  the  lines. 


the  absolute  value  of 

equal  to  the  ta 
It  follows  that 

(i.)  The  lines  are  parallel,  if  —  =  t,') 

(ii.)  The  lines  are  perpendicular,  if  aa' -^-bh' =  0. 

Ex.  1.  Write  down  the  equation  of  the  straight  line  through  (1,  2) 
perpendicular  to  x-y  =  Q. 

2.  Find  the  angles  between  the  lines 

x-2y  +  \=0\ 

x  +  Zy  +  2  =  0J 
and  .  4a;  +  Sy 

^x 
and  draw  the  lines. 

3.  Write  down  the  equation  of  the  straight  line  through  [a,  h)  per- 
pendicular to  hx-ay  =  a^  +  b\ 

4.  Write  down  the  equation  of  the  line  bisecting  the  line  joining  (1,  2), 
(3,  4)  at  right  angles,  and  the  equations  of  the  perpendiculars  upon  both 
lines  from  the  origin. 


+sy=m 

+  4y=l2l 


OF   THE  STRAIGHT   LINE 


13 


5.  Prove  that  l{x-a)  +  m{y-h)  =  0  is  the  equation  of  the  line  through 
{a,  b)  parallel  tolx  +  my  =  0;  and  that  m{x-a)-l{y -b)  =  0  is  the  equation  of 
the  line  through  {a,  b)  perpendicular  to  Ix  +  my-O. 

6.  Write  down  the  equations  of  the  lines  through  the  C.G.  of  the 
triangle  whose  angular  points  are  at  (4,  -5),  (5,  -6),  (3,  1)  parallel  and 
perpendicular  to  the  sides. 

§  10.  The  Length  of  the  Perpendicular  from  a  Point  (x^,  y^) 
upon  a  Straight  Line  whose  Equation  is  given.  "^ 

(i.)  If  the  equation  of 
the  straight  line  is  given 
in  the  "  perpendicular " 
form 

a;  cos  a  +  y  sin  a  =  j?,     (1) 

the  line  through  Vix^,  y^ 
parallel     to     it    is    given 

by 

{x  -  x^  COS  a  +  (y  -  y^)  sin  a  =  0, 
that  is,  by  x  cos  a  +  ?/  sin  a  =  ^Cq  cos  a  +  ^o  ^^^^  "• 

But  if  j?o  is  the  perpendicular  ON^  from  O  upon  the  line  (2), 
and  if  N,  Nq  are  on  the  same  side  of  0,  the  equation  of  PNq  may- 
be written  x  cos  a  +  ^  sin  a  =p^ . 
Since  (a^^,  y^  lies  upon  PNq,  we  have 

«Q  cos  a  +  ?/o  sin  a  =^Q . 
Also  the  perpendicular  from  ^{x^,  yo)  upon  the  line  (1)  is 
*ONo-ON,     (cf.  Fig.  6) 
i.e.  Pq  -p, 

i.e.  Xq  cos  a  +  y^  sin  a -p. 

In  the  case  when  Nq  lies  between  O  and  N,  we  have  to  take 

and  when  N,  N^  lie  on  opposite  sides  of  0,  ONq  makes  an  angle 
{a  +  tt)  with  Ox,  and  we  have  to  take 

*This  section,  and  the  examples  in  which  it  is  required,  may  be  omitted  by 
those  who  only  require  such  a  knowledge  of  analytical  geometry  as  is  necessary 
for  the  pages  of  this  book  referring  to  the  Calculus. 


U  THE  ANALYTICAL  GEOMETRY 

In  both  these  cases  the  length  of  the  perpendicular  is  given  by 
-  Xq  cos  a  -  y^  sin  a  +p. 
(ii.)  If  the  equation  of  the  line  is  given  as 

ax  +  hy  =  c     (c>  0),  (1) 

we  have  first  to  throw  this  into  the  "perpendicular"  form. 
Suppose  it  becomes 

X  cos  a  +  ^  sin  a  =p.  (2) 

Then,  by  equating  the  values  we  find  from  these  two  equa- 
tions for  the  intercepts  upon  the  axes,  we  obtain 


cos  a 

a 

sm  a    p 

h    ~c 

Therefore 

CCOSa 
csina 

and 

C2 

=^{a'  +  y^)f. 

where  there  is 
are  positive. 

Hence 

.'.    c 
no  ambiguity  in 

cos  a  =  - 

the  square  root, 
a 

as 

both 

P 

and 

c 

sma  = 


h 


and  p  =    ■ 

Therefore  the  "  perpendicular  "  form  of  the  line 

ax  +  by  =  c     (c  >  0) 

ax  by  c 

IS  ■  +        ^      - 


^a^  +  b^     s/a^  +  b^     s/a^  +  b^' 
Hence  the  length  of  the  perpendicular  from  (Xq,  y^)  upon 
ax  +  by-c  =  0 

is  +  (^^^o  +  H-g\ 

~\    Ja'^  +  b-'    J 

And  the  positive  sign  is  taken  when  (x^,  y^)  is  upon  the  opposite 


OF  THE  STEAIGHT  LINE  15 

side  of  the  line  from  the  origin,  the  negative  sign  when  it  is  on 
the  same  side  of  the  line  as  the  origin."^ 

This  result  holds  for  the  equation  of  the  straight  line,  in 
whatever  form  it  is  given.  The  reason  for  the  change  of  sign 
in  the  expression  for  the  length  of  the  perpendicular  is  that  the 
line  lx-\-mp  +  n  =  0  divides  the  plane  of  xy  into  two  parts.  In 
one  of  these  the  expression  Ix  +  my  +  n  is  positive ;  and  in  the 
other  it  is  negative.     Upon  the  line  the  expression  vanishes. 

Ex.  1.    Transform  the  equations 

{i.)  Sx±4y=^5,  {u.)Sx±4y=~5 

into  the  perpendicular  form,  and  write  down  the  vahie  of  a  for  each  with 
the  help  of  the  Trigonometrical  Tables. 

2.  Write  down  the  length  of  the  perpendicular  from  the  origin  upon 
the  line  joining  (2,  3),  (6,  7).  /  ,  "^ 

3.  Write  down  the  length  of  the  perpendicular  from  the  point  (2,  3) 
upon  the  lines       4x  +  3y  =  7,     5x  +  12y  =  20,     3x  +  4:y  =  S. 

4.  Find  the  inscribed  and  escribed  centres  of  the  triangle  whose  sides 
are  3x  +  4y=0,     5x-l2y=^0,    y  =  l5, 

and  the  equations  of  the  internal  and  external  bisectors  of  the  angles  of 
this  triangle,  distinguishing  the  different  lines. 

[A  fuller  discussion  of  the  subject  matter  of  this  chapter  is  given  in 
such  books  on  Analytical  Geometry  as  (i. )  Briggs  and  Bryan's  Elements  of 
Co-ordinate  Geometry,  Part  I. ,  Chapters  i.  -x. ;  (ii. )  Loney's  Co-ordinate 
Geometry,  Chapters  i.-vi. ;  (iii.)  C.  Smith's  Elejiientary  Treatise  on 
Conic  Sections,  Chapters  i.  and  ii. ;  and  (iv.)  Gibson  and  Pinkerton's 
Elements  of  Analytical  Geometry,  Chapters  i.-v. 

In  all  these  books  a  large  number  of  examples  will  be  found  illustrating 
the  points  we  have  discussed.] 

EXAMPLES  ON  CHAPTER  I 

1.    Find  the  equation  of  the  locus  of  the  point  P  which  moves  so  that 
(i.)  AP2  +  PB2  =  c2 
(ii.)  AP2-PB2  =  c^ 
(iii.)      AP.PB  =  c2, 
A  and  B  being  the  points  (  -  a,  0),  (a,  0). 

*  Rule. — To  find  the  length  of  the  perpendicular  from  a  given  point  (xo,  2/o) 
upon  a  given  straight  line  ^^ + ^^^Z  +  n = 0, 

insert  the  values  [x^,  ?/o)  in  place  of  {x,  y)  in  the  linear  expression  and  divide  by 
the  square  root  of  the  sum  of  the  squares  of  the  coefficients  of  x  and  y  in  this 
expression.     The  absolute  value  of 

IxQ+myo  +  n 

is  the  length  of  the  perpendicular. 


16  GEOMETEY  OF  THE  STEAIGHT  LINE 

2.  Find  the  equation  of  the  straight  line  through  ( -  1,  3),  (3,  2),  and 
show  that  it  passes  through  (11,  0). 

3.  Show  that  the  lines         Sx-   2i/  +  7  =  0 

4x+     2/  +  3  =  0 
iar  +  13y       =0 
-all  pass  through  one  point,  and  find  its  co-ordinates. 

4.  Find  the  equations  of  the  lines  through  the  origin  parallel  and  per- 
pendicular to  the  lines  of  Ex.  3  ;  also  those  through  the  point  (2,  2). 

5.  Find  the  equation  of  the  line  joining  the  feet  of  the  perpendiculars 
from  the  origin  upon  the  lines 

4x+  y=\l 
a:  -f-  2?/  —  5. 
•  6.    Draw  the  lines  4:y  +  2x  =  \2 

3y  +  4x  =  24. 
Find  the  equations  of  the  bisectors  of  the  angles  between  them,  dis- 
tinguishing the  two  lines. 

7.  The  sides  of  a  triangle  are 

X-    y+    \=0 
x-4y+   7  =  0 
x  +  2y-\l^0. 
Find         (i.)  the  co-ordinates  of  its  angular  points, 
(ii. )  the  tangents  of  its  angles, 

(iii.)  the  equations  of  the  internal  and  external  bisectors  of  these 
angles. 

8.  The  angular  points  of  a  triangle  are  at  (0,  0),  (2,  4),  (  -  6,  8).     Find 

(i.)  the  equations  of  the  sides, 
(ii. )  the  tangents  of  the  angles, 
(iii.)  the  equations  of  the  medians, 
(iv.)  the  equations   and   lengths   of   the   perpendiculars   from   the 

angular  points  on  the  opposite  sides, 
(v. )  the  equations  of  the  lines  through  the  angular  points  parallel 

to  the  opposite  sides, 
(vi.)  the  co-ordinates  of  the  C.G., 

(vii.)  the  co-ordinates  of  the  centres  of  the  inscribed,  circumscribed 
and  nine-points  circles. 

9.  Prove  that  the  area  of  a  triangle  whose  angular  points  are  the  origin, 
{x-^,  2/i)  and  (a^a,  3/2)5  i^  equal  to  the  absolute  value  of  ^i2/2-.V2^i .    g^^^j 

find  the  corresponding  expression  for  the  area  when  the  angular  points 
are  {x^,  y^),  {xo,  y.^)  and  (^3,  y.^. 

10.  Find  the  areas  of  the  triangles  given  in  Exs.  7  and  8. 


CHAPTER  II 

THE   MEANING   OF  DIFFERENTIATION 

§11.   The  Idea  of  a  Function. 

If  two  variable  quantities  are  related  to  one  another  in  such 
a  way  that  to  each  value  of  the  one  corresponds  a  definite 
value  of  the  other,  the  one  is  said  to  be  a  function  of  the  other. 
The  variables  being  x  and  ?/,  we  express  this  by  the  equation 
y=f{x).  In  this  case  x  and  y  are  called  the  independeilt  and 
dependent  variables  respectively.  Analytical  Greometry  furnishes 
us  with  a  representation  of  such  functions  of  great  use  in  the 
experimental  sciences.  The  variables  are  taken  as  the  co- 
ordinates of  a  point,  and  the  curve,  whose  equation  is 

gives  us  a  picture  of  the  way  in  which  the  variables  change. 

In  these  chapters  we  shall  assume  that  the  equation  y=f(x) 
gives  us  a  curve.  There  are,  however,  some  peculiar  functions 
which  cannot  thus  be  represented. 

§  12.   Examples  from  Physics  and  Dynamics. 

If  a  quantity  of  a  perfect  gas  is  contained  in  a  cylinder 
closed  by  a  piston,  the  volume  of  the  gas  will  alter  with  the 
pressure  upon  the  piston.  Boyle's  Law  expresses  the  relation- 
ship between  the  pressure  p  upon  unit  area  of  the  piston, 
and  the  volume  v  of  the  gas,  when  the  temperature  remains 
unaltered.     This  law  is  given  by  the  equation 

where  p^,  Vq  are  two  corresponding  values  of  the  pressure  and 
c.c.  B 


18  THE  MEANING  OF  DIFFERENTIATION 

the  volume.     When  the  volume  v  for  unit  pressure  is  unity^ 
this  equation  becomes 

and  the  rectangular  hyperbola,  whose  equation  is 

will  show  more  clearly  than  any  table  of  numerical  values  of  p 
and  V  the  way  in  which  these  quantities  change. 

When  the  pressure  is  increased  past  a  certain  point  Boyle's 
Law  ceases  to  hold,  and  the  relation  between  p  and  v  in  such  a 
case  is  given  by  van  der  Waals's  equation  :— 


(^  +  ^)(^-^)  =  ^' 


a,  h  and  c  being  certain  positive  quantities  which  have  been 
approximately  determined  by  experiment  for  different  gases. 
Inserting  the  values  of  a,  h  and  c  for  the  gas  under  consideration, 
and  drawing  the  curve 

(^  +  ^)(^-^^)  =  ^' 

with  suitable  scales  for  x  and  y,  the  way  in  which  p  and  v  vary 
is  made  evident. 

Such  illustrations  could  be  indefinitely  multiplied.  We  add 
only  two,  taken  from  the  case  of  the  motion  of  a  particle  in 
a  straight  line. 

When  the  velocity  is  constant,  the  distance  s  from  a  fixed  point 
in  the  line  to  the  position  of  the  particle  at  time  t  is  given  by 

S  =  Vt  +  SQ, 

where  s^  is  the  distance  to  the  initial  position  of  the  particle,, 
and  V  is  the  constant  velocity. 

The  straight  line  s  —  vt  +  Sq 

represents  the  relation  between  s  and  /,  the  co-ordinates  now 
being  referred  to  axes  of  s  and  t. 

When  the  acceleration  is  constant,  the  corresponding  equation  is 

s=yt^  +  v^t+SQ, 

where  /=  the  acceleration, 

t^Q  =  the  initial  velocity 
and  Sq  =  the  distance  to  the  initial  position. 


THE   MEANING  OF  DIFFERENTIATION  19 

In  this  case  we  have  the  parabola 

in  the  s,  ^  diagram. 

Also  in  both  these  cases  we  might  obtain  an  approximate  value 
of  s  for  a  given  value  of  t,  or  an  approximate  value  of  t  for  a 
given  value  of  s,  by  simple  measurements  in  the  figures  repre- 
senting the  respective  curves. 

§  13.    The  Fundamental  Problem  of  the  Differential  Calculus. 

The  aim  of  the  Differential  Calculus  is  the  investigation  of' 
the  rate  at  which  one  variable  quantity  changes  with  regard  to 
another,  when  the  change  in  the  one  depends  upon  the  change 
in  the  other,  and 'the  magnitudes  vary  in  a  continuous  manner. 
Of  course  there  are  also  cases  in  which  the  variable  we  are 
examining  depends  upon  more  than  one  variable.  However,  to 
such  cases  only  a  passing  reference  can  be  made  in  this  book.       ^ 

The  element  of  time  does  not  necessarily  enter  into  the  idea  of") 
a  rate,  and  we  may  be  concerned  with  the  rate  at  which  the 
pressure  of  a  gas  changes  with  the  \'olume,  or  the  length  of  a 
metal  rod  with  the  temperature,  or  the  temperature  of  a  con- 
ducting wire  with  the  strength  of  the  electric  current  along  it, 
or  the  boiling  point  of  a  liquid  with  the  barometric  pressure,  or 
the  velocity  of  a  wave  with  the  density  of  the  medium,  or  the 
cost  of  production  of  an  article  with  the  number  produced,  etc^ 
etc.  The  simplest  cases  of  rates  of  change  are,  however,  those  in 
which  time  does  enter,  and  we  shall  begin  our  consideration  of 
the  subject  with  such  examples. 

§  14.   Eectilinear  Motion. 

In  elementary  dynamics  the  velocity  of  a  point,  which  is  i 
moving  uniformly,  is  defined  as  its  rate  of  change  of  position, 
and  this  is  equal  to  the  quotient  obtained  by  dividing  the 
distance  traversed  in  any  period  by  the  duration  of  the  period, 
the  distance  being  expressed  in  terms  of  a  unit  of  length,  and 
the  period  in  terms  of  some  unit  of  time. 

When  equal  distances  are  covered  in  equal  times  this  fraction 
is  a  perfectly  definite  one  and  does  not  depend  upon  the  time, 
but  when  the  rate  of  change  of  position  is  gradually  altering, 
as,  for  instance,  in  the  case  of  a  body  falling  under  gravity,  the 


20  THE   MEANING  OF  DIFFERENTIATION 

value  of  such  a  fraction  alters  with  the  length  of  the  time  con- 
sidered. If,  however,  we  note  the  distance  travelled  in  different 
intervals  measured  from  the  time  /,  such  intervals  being  taken 
smaller  and  smaller,  we  find  that  the  values  we  obtain  for  what  we 
might  call  the  average  velocity  in  these  intervals  are  getting 
nearer  and  nearer  to  a  definite  quantity. 

For  example,  in  the  case  of  a  body  falling  from  rest  the 
distance  fallen  and  the  time  are  connected  by  the  follo\\'ing 
equation,  s  =  lgt'^. 

Let  us  fix  upon  a  certain  time  t  and  the  distance  s  which 
corresponds  to  that  time. 

Let  (s  +  hs)  be  the  distance  which  corresponds  to  the  time  {t  +  U). 

These  quantities  Ss  and  U  added  to  s  and  t  are  called  the 
"increments"  of  these  variables."^ 

Then         s-\-Ss  =  \g{t-\-  Uf  =  hgf^  +  gt  (St)  +  ^g  {8t)l 

Ss 
Therefore  ^/  ^  ^^  +  i^  (^^)' 

Now  let  t  be  kept  fixed,  but  let  the  increment  St  get  smaller 
and  smaller. 

It  is  clear  that  as  8t  tends  to  zero,  the  average  velocity 

gt  +  hg{8t\ 

for  the  interval  ^  to  (^  +  U),  approaches  nearer  and  nearer  to  the 
value  gt. 

This  value  towards  which  the  average  velocity  tends  as  the 
interval  diminishes  is  called  the  velocity  at  the  instant  t,  on  the 
understanding  that  we  can  get  an  "average  velocity"  as  near 
this  as  we  please  by  taking  the  interval  sufficiently  small. 

The  actual  motion  with  these  average  velocities  in  the  successive 
intervals  would  be  a  closer  and  closer  approximation  to  the  con- 
tinually changing  motion  in  proportion  to  the  minuteness  of  the 
subdivisions  of  the  time.  The  advantage  of  the  method  of  the 
Differential  Calculus  is  that  it  gives  us  a  means  of  getting 
these  "instantaneous  velocities,"  or  rates  of  change,  at  the  time 
considered.     When   the   mathematical   formula   connecting    the 

*  When  these  *  ^increments''  are  small,  it  is  convenient  to  speak  of  them  as  "  the 
little  piece  added  to  s  "  and  "  the  little  piece  added  to  t."  It  has  to  be  noticed  that 
the  symbols  ds  and  5t  have  to  be  taken  as  a  whole.  The  beginner  is  apt  to  look  upon 
Ss  as  dxs,  when  he  uses  it  in  an  algebraical  expression. 


THE   MEANING  OF  DIFFEEENTIATION  21 

quantities  is  given,  we  can  state  what  the  rate  of  change  of 
the  one  is  with  regard  to  the  other,  without  being  dependent 
upon  an  approximation  obtained  by  a  set  of  observations  in 
gradually  diminishing  intervals. 

§  15.   Limits.     Differential  Coefficient.  — ^ 

If  a  variable  which  changes  according  to  some  law  can  be 
made  to  approach  some  fixed  constant  value  as  nearly  as  we  please, 
but  can  never  become  exactly  equal  to  it,  the  constant  is  calledj 
the  limit  of  the  variable  under  these  circumstances.  Now  if 
this  variable  is  x,  and  the  limiting  value  of  x  is  a,  the  dependent 
variable  y  (where  y=f{x))  may  become  more  and  more  nearly 
equal  to  some  fixed  constant  value  J  as  x  tends  to  its  limit  a, 
and  we  may  be  able  to  make  y  differ  from  h  by  as  little  as  we 
please,  by  making  x  get  nearer  and  nearer  to  a.  In  this  case 
h  is  called  the  limit  of  the  function  as  x  approaches  its  limit  a,  or 
more  shortly,  the  limit  of  the  function  for  x  =  a. 

As  the  variable  x  is  only  supposed  gradually  to  tend  towards 
the  value  a,  without  actually  attaining  that  value,  it  is  better  to 
write  this  in  the  form  j^^  iy\  ^  j 


rather  than  in  the  form 


Lt  {y)^h. 


In  this  way  we  emphasize  the  fact  that  it  is  not  the  value  of  y 
for  X  equal  to  a  with  which  we  are  dealing.  What  we  are 
concerned  with  is  the  limiting  value  of  ?/  as  a;  converges  to  a  as 
its  limit. 

Ex.    (i.)  If  2/=^' 

Lt(2/)-l. 

(ii.)  If  y^xlogioX, 

Lt(y)  =  0. 

(iii.)  If  2/  =  i, 

Lt  (y)  =  00  , 

X— >0 

or,  more  correctly,  y  has  no  limit  as  x  tends  to  zero.* 

*  For  a  fuller  elementary  discussion  of  the  idea  of  a  limit,  see  Love's  Elements 
of  the  Differential  and  Integral  Calculus,  Ch.  II.,  §§  19,  20  and  Appendix. 

The  subject  is  also  discussed  in  such  standard  text-books  as  Lamb's,  Gibson's, 
and  Osgood's. 


22  THE  MEANING  OF  DIFFERENTIATION 

In  this  last  example  the  function  increases  without  limit  as  -x 
approaches  its  limit.  We  might  have  the  corresponding  case 
of  X  increasing  without  limit  and  the  function  having  a  definite 
limit :  e.g.  if  y^^.  ^^ere  0  <  a  <  1, 

Lt  {y)  =  0. 

This  idea  of  a  limit  has  already  (§  14)  been  employed,  and 
when  s  =  ^gf,  the  velocity  at  the  time  t  of  the  moving  point  is 
what  we  now  denote  by  the  symbol 

In  the  general  case,  when  the  relation  between  s  and  t  is 
s=f(^t),  we  take  the  distance  at  the  time  {t  +  8t)  as  (s  +  Ss). 
Then  we  have  s  +  8s  =f{t  +  8t) 

and  •  ^s_f(t  +  8t)-f{t)^    ■ 

8t  8t 

Hence  the  velocity  at  the  time  t  is  given  by 


St-^o\^i 


'A       8t       /• 

The  limiting  value  of  the  ratio  of  the  increment  of  s  to  the 
increment  of  t,  as  the  increment  of  t  approaches  zero,  is  called  the 
differential   coefficient  of  s   with   regard   to   t.     Instead  of  wi'iting 

Lt  (^\  we  use  the  symbol  -y-  for  this  limiting  value. 
st-^\otJ  dt 

It  must,  however,  he  carefully  noticed  that  in  this  symbol  ds  and  dt 

cannot,  so  far  as  we  are  here  concerned,  he  taken  separately,  and  that 

—  stands  for  the  result  of  a  definite  mathematical  operation,  namely, 

the  evaluation  of  the  limiting  value  of  the  ratio  of  the  corresponding 
increments  of  s  and  t,  as  the  increment  of  t  converges  to  zero.^ 

We  shall  see  later,  in  §  38,  that  there  is  another  notation  in 
which  ds  and  dt  are  spoken  of  as  separate  quantities,  but  until 
that  section  is  reached,  it  will  be  well  always  to  think  of  the 
differential  coefficient  as  the  result  of  the  operation  we  have  just 
described. 

*For  this  and  other  reasons  we  shall  often  write  -nf{t),  instead  of  ^^  . 
This  is  also  written  f'{t).  ^^  ** 


THE  MEANING  OF  DIFFERENTIATION  23 

It  is  clear  that  if  8t  is  very  small,  the  corresponding  increment 

ds 
of  .9,  namely  Ss,  will  be  very  approximately  given  by  —8t.     Still 

dt 

it  is  not  a  true  statement,  but  only  an  approximation,  to  say 

that  in  this  case  ^g 

8s  =  .7  St. 
dt 

However,  this  approximation  is  very  important.  It  may  be 
employed  in  finding  the  change  in  the  dependent  variable  due 
to  a  small  change  in  the  independent  variable,  or  the  error  in  the 
evaluation  of  a  function  due  to  a  small  error  in  the  determination 
of  the  variable,  provided  we  know  the  differential  coefficient 
of  the  function. 

We  add  some  examples  in  which  the  differential  coefficients 
are  to  be  obtained  from  the  above  definition,  viz. — 


If  s  = 

Ex.  1.    If  s=at  +  b 

2.  If  s  =  at'^  +  2ht-hc 

3.  If  d=ut, 

4.  If  y  =  mx  +  n 

5.  If  y  =  ax'^, 


§  16.  Geometrical  Illustration  of  the  Meaning  of  the  Diflfer- 
€ntial  Coefficient. 

The  gradient,  or  slope,  of  a  straight  line  has  been  defined 
in  §5.  The  gradient  of  a  curve  at  any  point  is  the  gradient 
of  the  tangent  at  that  point. 

We  obtain  another  illustration  of  the  meaning  of  the  differ- 
ential coefficient  by  considering  the  gradient,  or  slope,  of  the 
<5urve  y=f{x). 

Let  P  be  a  certain  point  (ic,  ?/),  which  we  suppose  fixed. 
Let  Q  be  another  point,  its  co-ordinates  being  denoted  by 
{z  +  8x,y  +  8y). 


^')'  'h^. 

(f(t  +  St)- 
A          8< 

-fim 

ds 

-c,   ^^=2{at  +  b). 

dd 
dt=''' 

dx 

f=2ax. 
dx 

24 


THE  MEANING  OF  DIFFERENTIATION 


Let  the  tangent  at  P  make  an  angle  <^  with  Ox. 
Then,  in  Fig.  7, 

0M  =  ;«        ^      and 

0^  =  x  +  8x\ 
M^  =  8x      J 

MP  = 

MQ  = 

-{y  +  8y)=f{x  +  8x)             \ 
=  8y          =f{x  +  8x)-fix).) 

y 

A 

/q 

< 

Thus  the  slope  of  the  secant 

PQ 

=  tanHPQ, 

^y 

~8x 

y 

/^>v 

H 

f{x  +  8x)-f(x) 
8x 

Now,  if  we  keep  P  fixed, 
and   let   Q   approach   P,  the 

M 


Fig. 


secant  PQ  gets  nearer  and 
nearer  the  tangent  at  P,  and 
the  limiting  value  of  the  frac- 


tion °y  as  8x  gets  smaller  and  smaller,  is  tan(^.* 


Thus,  with  the  same  notation  as  before. 


ill 
dx 


We  have  therefore  shown  that  when  the  dependent  and  independent 
variable  are  the  ordinate  and  abscissa  of  a  point  upon  a  curve,  the 
differential  coefficient  is  equal  to  ths  gradient  of  the  curve. 

Since  the  slope  of  the  tangent  is  known  when  -j-  is  found,  we  can 
write  down  the  equation  of  the  tangent  at  a  point  {x^,  y^)  on  the  curve 
y  =f{x),  when  the  value  of  -^  at  that  point  is  known. 

This  equation  is  [cf .  §  6,  (B)] 

- — —  =  the  value  of  -^  at  a^'  =  a;^ . 


*  The  increments  dx,  by  need  not  be  positive.  Unless  the  curve  has  a  sharp 
comer  at  the  point  considered,  the  limiting  position  of  the  secant  PQ  would  be 
the  same  whether  bx  were  positive  or  negative. 


THE   MEANING  OF  DIFFERENTIATION 


25 


Writing  f{p:)  for  the  differential  coefficient  of  f{x),  and  /'(^o) 
for  the  value  of  f'{x)  when  x  has  the  value  x^^  this  equation 
becomes  2/ "  ^o  =  (^  "  ^o)/ W- 

Ex.    Find  the  vahie  of  -^  at  the  point  (2,  1)  on  the  parabola  ^y  =  x^y 

and  show  that  the  equation  of  the  tangent  at  that  point  is 

x-y=l.     [Cp.  p.  83.] 

§17.  Approximate  Graphical  Determination  of  the  Differ- 
ential Coeificient. 

When  the  equation  connecting  x  and  y  is  such  that  the  curve 

may  be  easily  drawn,  the  slopes  of  the  various  positions  of  the 

secant  PQ,  as  Q  is  made  to  move  nearer  and  nearer  to  P,  will 

give  a  series  of  values  more  and  more  nearly  approximating  to 

(lij 
the  value  of  -j-  at  that  point.     An  instructive  example  is  the 

case  of  the  curve 


^j  =  x^, 


8y 


in  which  the  following  table  of  values  of  8x,  8y  and  ^  can  readily 

8y  ^ 

be  obtained.      The   way  in   which   ^  approaches  its   limiting- 
value  2  at  the  point  where  a;  =  1  is  evident. 


6x 
Sx 

1 
3 

3 

•9 

•8 

•7 

•6 

•5 

•4 

•3 

•2 

•1 

•09 

•08 

•07 

•06 

•05 

•04 

•03 

•02 

•01 

2-61 
2-9 

2-24 

2-8 

1-89 
2-7 

1-56 
2-6 

1-25 
2-5 

•96 
2-4 

•69 
2-8 

•44 
2-2 

•21 
2-1 

•1881 

•1664 

•1449 

•1236 

•1025 

•0816 

•0609 

•0404. 

•0201 

2^09 

2^08 

2-07 

2-06 

2^05 

2-04 

2-08 

2-02 

2-01 

§  18.   Repeated  Differentiation. 

We  have  now  seen  what  is  meant  by  the  differential  coefficient 
of  a  function  of  a  single  variable.  The  process  of  obtaining  the 
differential  coefficient  is  called  differentiating  the  function.  In 
the  chapters  which  immediately  follow  we  shall  show  how  to 
differentiate  the  most  important  functions,  and  we  shall  prove 
some  general  theorems  in  differentiation.  These  will  allow  us  to 
extend  very  widely  the  class  of  function  for  which  we  can  write 
down  the  differential  coefficients. 

It  is  immaterial  what  symbols  we  use  for  the  dependent  and 
independent  variables.      We   began   by  using  s  and  t  in   the 


I 


26  THE  MEANING  OF  DIFFERENTIATION 

dynamical  illustration  of  a  rate  of  change.  Then  we  used  the 
relation  y=f{x),  and  found  that  the  differential  coefficient  of  y 
with  regard  to  x  was  the  slope  of  the  curve  y  =f(x)  at  the  point 
{x,  y).  We  shall  use  this  geometrical  notation  most  frequently, 
since  one  of  the  best  introductions  to  the  Calculus  is  through  its 
applications  in  Analytical  Geometry. 

The  differential  coefficient  -/,  or  f'(x),  is  itself  a  function  of  x, 

and  we  may  differentiate  this  function.     Its  differential  coefficient 

is  written  -—,  or  f'''{x),  and  is  called  "the  second  differential 

coefficient "  of  y,  or  of  f(x). 

This  process  may  be  repeated  indefinitely.  The  differential 
coefficient  of  the  second  differential  coefficient  being  called  the 

third  differential  coefficient,  and  being  written  J,  or  f"{x),  etc. 

7  cix 

From  this   point  of   view  -f-,   or  f'(x),  is  called    "the   first 

cix 

differential  coefficient." 

Consider  the  case  y  =  mx  +  n. 

We  know  from  Chapter  I.  that  this  is  the  equation  of  a 
straight  line  of  gradient  m. 

Therefore  we  have  3-  =  ^• 

ax 

Also  as  the  gradient  m  is  the  same  for  all  values  of  x,  its  rate 
of  change  is  zero. 

Therefore  cW^^' 

Again  take  the  case  y  =  x'^. 

We  have  already  seen  how  to  differentiate  such  a  function 
(cf.  §§  14,  15)  proceeding  from  the  definition  of  the  differential 
coefficient.  Later  we  shall  obtain  a  rule,  which  will  enable  us  to 
write  down  the  answer  immediately. 

With  the  method  already  employed,  we  begin  with  the  value 
x,  and  we  have  y  ^  ^2^ 

Then  we  take  Sx  for  the  increment  of  x,  and  we  write  8y  for 
the  corresponding  increment  of  y. 

Therefore  we  have       y  +  Sy  =  {x  +  Sx)^. 


THE  MEANING  OF  DIFFERENTIATION 


27 


From  these  two  equations  it  follows  that 


Bx 


=  2x  +  {8x). 


Lt 


.(2)-" 


Thus 


dx 


=  2x. 


To  find  the  second  differential  coefficient,  we  have  to  differ- 
entiate the  expression  for  -^. 

In   this   case   we   find   at  once   by  calculation,  or  from  our 
knowledge  of  the  graph  of  2x,  that 


From  Fig.  8,  it  is  obvious  that,  when  -^  is  positive,  the  tangent  is 

ax 

inclined  at  an  acute  angle  to  the  axis  of  x :  that,  when  it  is  negative, 

this  angle  is  oMuse.     A  positive  -j-  means  that  y  increases  with  x  at 

dy 
that  point:  a  negative  -^  means  that  y  diminishes  as  x  increases. 

dv 
When  -j-  vanishes,  the  tangent  must  be  parallel  to  the  axis  of  x. 

Let  us  imagine  the  curve  ABC...  to  stand  for  a  road,  and  that 
a  traveller  is  marching  along  it  in  the  positive  direction  of  the 


28 


THE  MEANING  OF  DIFFERENTIATION 


axis  of   X,   which  is  horizontal.      When   the   traveller  ascends, 
-  is  positive  :   when  he  descends,  -^  is  negative ;    and  if  the 


dx 


dx 


road  is  rounded  off  and  no  sudden  changes  of  gradient  occur, 
when  he  ceases  to  ascend  and  begins  to  descend,  or  the  reverse, 

-^  changes  sign  by  passing  through  zero.     [See  also  p.  71.] 

What  infoi'mation    can  we    obtain  from    the   second    differential 
coefficient  of  y  regarding  the  curve  y  =f{x)  ? 

We  have  seen  that  J  stands  for  the  rate  of  change  of  the 
ax'^ 

gradient.     It  follows  that  along  the  parts  of  the  curve  where 

the  gradient  is  increasing,  -^  is  positive.     Also  that  along  the 

ax 

parts  of  the  curve  where  the  gradient  is  diminishing,  J  is 
five.  '^'■^ 


.^.>o 


Fig.  9. 

This  can  also  be  put  in  the  following   way  :    — ^  is  positive^ 

ax 

when  the   curve  y=f(x)  is  concave  upwards;    that  is,   concave, 

when  looked  at  from  above.     Also  ~  is  negative,  when  the  curve 

ax 

is  convex  upwards ;  that  is,  convex,  when  looked  at  from  above. 

The  second  differential  coefficient  has  also  an  impmiant  applicatioih 

in  Dynamics. 


THE   MEANING  OF  DIFFERENTIATION  29 

The  acceleration  of  a  moving  point  is  defined  as  its  rate  of 
change  of  velocity. 

Let  us  return  to  the  case  of  the  motion  of  a  point  along  a 
straight  line,  in  which  the  distance  and  the  time  are  connected 
by  the  relation  s  =  f(t). 

Let  the  velocity  at  the  time  t  be  ^^ 

Then  we  know  that        '^  =  77  =/'(0' 

Also  the  acceleration  at  the  time  t  is  the  rate  of  change  of  the 
velocity  at  that  time. 

Thus  the  acceleration  =  — 

dt 

E.g.     If  •  s  =  \gt\ 

ds 

d^s 
and  the  acceleration  =  -j-z=g, 

cct 

Again,  when  the  velocity  is  increasing,  the  acceleration  is 
positive ;    when    the   velocity   is    decreasing,   the   acceleration   is 

negative. 

...         d^s 
Therefore  the  sign  of  the  second  differential  coefficient,  -j-^, 

tells  us  whether  the  velocity  is  increasing  or  decreasing  at  the 
instant  considered.     We  shall  return  to  this  question  later,  and 

d^s 
we   shall   see  that  when  the  second  differential  coefficient  -^ 

vanishes  for  a  certain  value  of  t,  and  is  positive  just  before  that 
value  of  t,  and  negative  just  after  it,  then  at  that  particular 
instant  the  velocity  has  a  maximum  value.  Also  that  when 
the  change  of  sign  is  from  negative  to  positive,  the  velocity 
has  a  minimum  value  at  that  time.     [Cf.  §  38.] 

EXAMPLES  ON  CHAPTER  II 

The  differential  coefficients  required  in  the  examples  on  this  chapter  are 
to  be  obtained  from  the  definition. 

1.    Plot  the  curves        (i.)  y  =  x  +  x^        (ii.)  y  =  ofi, 
and  show  that  they  have  the  same  gradient  when  x=l. 


30  THE  MEANING  OF  DIFFERENTIATION 

2.  By  considering  the  area  of  a  square  and  the  volume  of  a  cube,  show- 
that  the  differential  coefficients  of  x'^  and  x^  are  2x  and  Zx"^  respectively. 

3.  Show  that  the  curves  y  =  x^  and  y  =  x^  intersect  at  the  origin  and 
the  points  (1,  1),  (-1,  1),  and  that  at  each  of  the  two  latter  points  the 

angle  between  the  tangents  is  tan"^  -. 

y 

4.  Show  that  the  gradient  of  the  curve  y  =  x^  -  Zx  at  the  point  where 
x  =  2  is  9.     Find  the  equation  of  the  tangent  there  and  trace  the  curve. 

5.  Find  where  the  ordinate  of  the  curve  y  =  Zx-4x'^  increases  at  the 
same  rate  as  the  abscissa,  and  where  it  decreases  five  times  as  fast  as  the 
abscissa  increases. 

6.  If  8  =  nt-\gt'^,  find  the  values  of  the  velocity  and  acceleration  at 
the  time  t. 

7.  A  cylinder  has  a  height  h  ins.  and  a  radius  r  ins. ;  there  is  a  possible 
small  error  br  in  r.  Find  an  approximate  value  of  the  possible  error  in 
the  computed  volume. 

8.  Find  approximately  the  error  made  in  the  volume  of  a  sphere  by 
making  a  small  error  5r  in  the  radius  r.  The  radius  is  said  to  be  20  ins. ; 
give  approximate  values  of  the  errors  made  in  the  computed  surface  and 
volume  if  there  be  an  error  of  "1  in.  in  the  length  assigned  to  the  radius. 

9.  The  area  of  a  circular  plate  is  expanding  by  heat.  When  the  radius 
passes  through  the  value  2  ins.  it  is  increasing  at  the  rate  of  -01  in.  per  sec. 
Show  that  the  area  is  increasing  at  the  rate  of  "04^  sq.  in.  per  sec.  at  that 
time. 

10.  The  length  of  a  bar  at  temperature  0°  is  unity.  At  temperature  f 
its  length  I  is  given  by  the  equation 

l^l+at  +  W^', 
find  the  rate  at  which  the  bar  increases  in  length  at  temperature  f,  and 
give  an  approximation  to  the  increase  in  length  due  to  a  small  rise  in 
temperature, 

11.  If  the  diameter  of  a  spherical  soap-bubble  increases  uniformly  at 
the  rate  of  '1  centimetre  per  second,  show  that  the  volume  is  increasing 
at  the  rate  of  '2ir  cub.  cent,  per  second  when  the  diameter  becomes 
2  centimetres. 

12.  A  ladder  24  feet  long  is  leaning  against  a  vertical  wall.  The  foot 
of  the  ladder  is  moved  away  from  the  wall,  along  the  horizontal  surface  of 
the  ground  and  in  a  direction  at  right  angles  to  the  wall,  at  a  uniform 
rate  of  1  foot  per  second.  Find  the  rate  at  which  the  top  of  the  ladder  is 
descending  on  the  wall,  when  the  foot  is  12  feet  from  the  wall. 


CHAPTER  III 

DIFFERENTIATION   OF  ALGEBRAIC   FUNCTIONS;    AND   SOME 
GENERAL  THEOREMS   ON   DIFFERENTIATION 

§  19.   The  Differentiation  of  x^  when  n  is  a  Positive  Integer. 
We  have  already  seen  that, 

when     y  =  x'^, 

dx 
A  similar  argument  would  show  us  that, 
when     y  =  x^, 


and  that,  when 


ax 


dx 


These  suggest  that 

when     y  =  x^, 


dy 


nx 


n-l 


dx 

As  a  matter  of  fact  this  formula  is  true,  when  n  is  any  number 
independent  of  x.  However  we  shall  prove  it,  at  present,  only 
for  the  case  oi  n  a  positive  integer.  The  cases  when  the  index  of 
the  power  of  ic  is  a  fraction  or  negative  we  shall  examine  later."^ 

*  In  the  first  edition  of  this  book  the  usual  proof  of  this  theorem  is  given, 
the  Binomial  Theorem  for  any  index  being  employed.  The  student,  who  under- 
stands the  use  of  Infinite  Series,  will  probably  prefer  that  proof,  but  it  seems 
better  to  give  those  who  have  not  read  that  difficult  portion  of  Algebra,  or  have 
not  properly  understood  it,  an  alternative  method.  Similar  changes  are  made 
in  the  discussion  of  the  differentiation  of  the  exponential  and  logarithmic 
functions,  and  our  subject  is  developed  without  the  use  of  the  Theory  of  Infinite 
Series  at  all.     The  proofs  referred  to  are  given  in  the  Appendix  (p.  129). 


32    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 


As  usual,  we  begin  with  the  value  x^  and  we  put 

y  =  %\  (1) 

Then  we  take  the  increment  ^x  of  x^  and  we  write  hj  for  the 
corresponding  increment  of  y. 

It  follows  that  y  +  ^  =  {x^  Sxf. 

Now  we  know  from  Elementary  Algebra  that 

{a  +  hf  =  ft"  +  na^-'  b  +  ^-^)  ^-2^2  + 

when  71  is  a  positive  integer. 
Therefore  we  have 


+  5", 


y^^  =  x^  +  nx^'-' {8x)  +  ^^^  ^^^ ««-2(aa^)2  +  . . .  +  (6a;)' 


(2). 


From  (1)  and  (2)  we  have 


8y  =  wic"~^  (Sx)  + 


/l(ll-  1)      „     o 


1.2 


a;'*--(S.7;)2+...+(5a:)". 


=  nx-'  +  'i:i?-J-^  x"--(3^)  +  . . .  +  (8xY-\ 


Therefore 
Sy 

8x'    ""^       '       1.2 

Now  7i  is  a  definite  positive  integer,  and  there  are  (^  -  1) 

terms  in  this  expression  after  the  first.     All  of  these  terms  have 

the  factor  8x.     If  we  let  8x  tend  to  zero,  the  sum  of  these  terms 

must  vanish  in  the  limit. 

^8y^ 


It  follows  that  Lt  (g|)  =  ^^"~'- 


Thus  we  have  proved  that^  when  n  is  a  positive  integer,  the  differ- 
ential coefficient  of  x""  is  naf~'^. 

Ex.    Fill  lip  the  blank  column  in  the  following  table  : 


fix). 

fix). 

X 

I 

.r2 

x' 

x^ 

L.    ,   ■ 

x^ 

x« 

x« 

x^ 

GENERAL  THEOKEMS  ON  DIFFERENTIATION      33 

§  20.   General  Theorems  on  Differentiation. 

Before  proceeding  to  obtain  the  differential  coefficients  of 
other  functions,  it  will  be  useful  to  show  that  many  complicated 
expressions  can  be  differentiated  by  means  of  this  result,  with 
the  help  of  the  following  general  theorems  : — 

Proposition  I.   Differentiation  of  a  Constant. 

It  is  clear  that,  if  y  =  a,  the  slope  of  the  line  is  zero,  and 

dv 

-/  =  0.     In  other  words,  it  is  obvious  that  if  a  magnitude  remains 

the  same  its  rate  of  change  is  zero. 

Thus  the  differential  coefficient  of  a  constant  is  zero. 

Proposition  II.  Differentiation  of  the  Product  of  a  Constant  and 
a  Function  of  x. 

Let  y  =  au,  where  a  is  a  constant,  and  u  is  a  function  of  x. 

We  begin  with  the  value  x,  and  we  take  5a:  for  the  increment 
of  x. 

When  x  becomes  x-\-^jX,  let  u  become  w  +  Sm,  and  y  become 
y  +  hy. 

Then  y  +  8y  =  a(u  +  8u), 

,  8y       8u 

and  /  =  a^. 

ox        ox 

For  the  value  of  x  considered,  we  are  supposed  to  know  that 

-T-  exists  :  in  other  words,  that 
dx 


is  a  definite  number. 


t.© 


It  follows  that  Lt  (^j 


Sx->0 


M        .    T^     /^^" 


=  a  Lt  (f). 


Therefore  -/  =  a  -^. 

ax       ax 

Thus  the  differential  coefficient  of  the  product  of  a  constant  and  a 
function  is  equal  to  the  product  of  the  constant  and  the  differential 
coefficient  of  the  function. 

The  geometrical  meaning  of  this  theorem  is  that  if  all  the 
ordinates  of  a  curve  are  increased  in  the  same  ratio,  the  slope  of 
the  curve  is  increased  in  the  same  ratio. 

The  dynamical  meaning  will  be  obvious  to  the  reader, 
c.c.  c 


34    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

Proposition  III.   Differentiation  of  a  Sum. 
Let  y  =  u  +  v. 

Then,  as  before,  y  +  8y  =  (u  +  Su)  +  (v  +  8v), 

and  %^S»^_ 

8x     8x     8x 

We  are  supposed  to  know  that,  for  the  value  of  x  considered^ 
^  and  ^  have  definite  limiting  values  as  8x^0. 

It  follows,  on  proceeding  to  the  limit,  that 
dy     du     dv 
dx     dx     dx 

The  same  argument  applies  to  the  sum  (or  difference)  of 
several  functions,  and  we  see  that  the  differential  coefficient  of  such 
a  sum  is  the  sum  of  the  several  differential  coefficients. 

Ex.    Differentiate  the  following  functions  : — 
(i.)  x{2^xf 
(ii.)  {a  +  hx  +  cx^)x 

/yt4  /v»a  /vtis 

(iii.)  -^  +  -^  +  ^  +  ^  +  ^ 
(iv.)  2  +  2x  +  3x2. 

Proposition  IV.  Differentiation  of  the  Product  of  Two  Functions. 

Let  y  =  uv. 

Then,  as  before,  y  +  8y  =  {u  +  8v)  {v  +  8v). 

Thus  8y  =  v8u  +  u  8v  +  8u  8v, 

8y       8u        8v     „    8v 
and  ^  =  v^  +  u^  +  8u^- 

ox  ox  ox  ox 

We  are  supposed  to  know  that,  for  the  value  of  x  considered, 

^  and  —  have  definite  limiting  values  as  8x  ->  0. 
ox  ox 

In  this  case  8u  ->  0,  as  8x  -^  0.      It  follows,  on  proceeding  to 
.the  limit,  that  dy_    du       dv 

dx~    dx       dx 


This  result  may  be  written 

I  dy _l  du     I  dv, 

y  dx~ u  dx     V  dx 


GENERAL  THEOREMS  ON  DIFFERENTIATION     35 

and  when  y  =  uvw^  we  would  obtain  in  the  same  way, 

\  dy _\  du     \  dv     1  dw      /pr  o  oi  \ 
y  dx    u  dx     V  dx    w  dx 

In  the  case  of  two  functions  it  is  easy  to  remember  that  the 

differential  coefficient  of  the  product  of  two  functions  is  equal  to  the  first 
function  x  the  differential  coefficient  of  the  second  +  the  second  function 
X  the  differential  coefficient  of  the  first. 

Ex.    Differentiate  the  following  functions  as  products  : — 
(i.)  (l+x^-) {2x^-1) 
(ii.)  (2a;2+l)(x  +  2)2 
(iii.)  {ax  +  h)^{cx  +  d)'^ 
(iv.)  x{x  +  l){x  +  2), 
and  show  that  the  results  are  the  same  if  the  expressions  are  multiplied 
out  and  then  differentiated. 

Proposition  V.     Differentiation  of  a 
Let  y  =  u/v. 

Then,  as  before,  y  +  8y  = ^, 

-  ^      u  +  8u     u     v8u-u< 

and  oy  = 


v  +  8v     V       „/^     8v 


Therefore 


8x 


('4) 

8u        8v 
8x       8x 


We  are  supposed  to  know  that,  for  the  value  of  x  considered, 

cT-  and  TT-  have  definite  limitinsr  values  as  8x  ->  0. 
8x         ox 

In  this  case  8v  ->  0,  as  8x-^0. 

Proceeding  to  the  limit,  it  follows  that 

du^       dv 
dy  _    dx       dx  * 

dx  ~        v^ 

*  This  result  may  be  obtained  by  writing 

vy=u, 
and  then  differentiating  both  sides  of  the  equation. 


36     DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

In  words,  to  find  the  differential  coefficient  of  a  quotient,  from 
the  product  of  the  denominator  and  the  differential  coefficient  of  the 
numerator  subtract  the  product  of  the  numeratm-  and  the  differential 
coefficient  of  the  denominator,  and  divide  the  result  by  the  square  of 
the  denominator. 

We  can  use  this  result  to  find  the  differential  coefficient  of  x"", 
when  n  is  a  negative  integer. 

Let  n=  -m,  where  m  is  a  positive  integer. 

Then  we  have  y  =  -^' 

Oil 

Therefore  -f-  = ^- — , 

dx         Q?"' 

since  the  differential  coefficient  of  the  numerator  is  zero,  and 
the  differential  coefficient  of  the  denominator  is  m.7f' ~\ 


Thus 


dx 


In  particular,  if  y=-> 

dx         x? 
We  return  to  this  on  p.  40. 

Ex.    Differentiate  the  following  expressions 

(i.)  pi  (ii.)  (^+1)1^)  (iii.) 


(iv  )  (^+^)'  (V  )  i±^'  {vi  )  ^^'  +  2&^  +  g 

These  five  formulae,  with  the  help  of  the  result  of  §  19,  enable 
us  to  differentiate  a  large  number  of  expressions,  but  they  do 
not  apply  directly  to  such  cases  as  {ax  +  />)^^,  (ax^  +  2bx  +  c)^^,  etc. 

Each  of  the  above  expressions  is  a  function  of  a  function  of 
X,  and  we  proceed  to  prove  another  general  theorem  : — 

Proposition  VI.     Differentiation  of  a  Function  of  a  Function. 
Let  y  =  F(i^), 

where  w  =/(.t) 

/e.g.  y  =  u^'^,    \ 

\where    u  =  a^  +  x^J. 


GENERAL  THEOREMS  ON  DIFFERENTIATION     37 

We  begin  with  the  value  x,  and  we  take  8x  for  the  increment 
of  X. 

Then  when  x  is  changed  to  a;  +  8x, 

let  u  become  u  +  8u, 

and  y  become  V  +  ^V'} 

the  functions  being  such  that  for  a  small  change  in  x  we  have 
a  definite  and  small  change  both  in  u  and  y. 

But  ^/  =  '/-T- 

OX     bu     ox 

Now  we  are  supposed  to  know  that,  for  the  value  of  x  con- 
sidered, —  has  a  definite  limiting  value  as  6a: ->0.     In  this  case, 

when  8a; ->0,  8m ->0.     Also  we  are  supposed  to  know  that  -^ 
has  a  definite  limiting  value  as  8w->0. 

It  follows,  on  proceeding  to  the  limit,  that  , 

dy  _  dy    du 
dx  ~  du    dx* 

It  is  important  to  notice  that  this  rule 

dy  _  dy    du 
dx  ~  du     dx 

cannot  be  inferred  from  striking  out  the  dv!^^  as  if  the  expressions 
were  fractions.     We  have  already  laid  stress  on  the  fact  that 

the   differential  coefficient  -Jf-   is  not   to   be  looked  upon  as  a 

fraction  dy  divided  by  dx. 

Corollary.     If  we  put  2/  =  a;  in  the  above  result,  we  obtain 

dx    du  _ 
du    dx~   ' 

It  follows  that  _-  =  __. 

du     du 

Tx  * 

Altering  this  notation,  we  can  say  that 

dy    dx     ,         J  ^1,  4.    dy      1 
3— X:5— =  1,    and  that    3— =  -5— • 
dx    dy  dx    dx 

d^ 


38    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

This  theorem  could  have  been  proved  directly,  using  an  argu- 
ment very  similar  to  the  above. 

Also  it  is  instructive  to  see  that  it  follows  immediately  from 
the  geometrical  interpretation  of  the  differential  coefficient. 

Ex.  1.    Differentiate  {x+l)*. 


Let 

y  =  {x  +  l)\ 

Then  we  have 

y  =  u\   where  u  =  x  +  l. 

But 

dy  _  dy    du 
dx  "  du    dx' 

It  follows  that 

dx 

=  4(x  +  l)3. 

2.  Differentiate  (2a; +  1)6. 

Let  y  =  {2x  +  \f. 

Then  we  have  y  =  u^,   where  u  =  2x+l. 

But  dy^dy^du^ 

dx    du    dx 

It  follows  that  ^ = 6w5  X  2 

dx 

=  12(2a;  +  l)5. 

3.  Differentiate  {l-'^xf^. 

Let  y^[\-^xy^ 

Then  we  have  y  =  u^^,   where  2i=l-3x. 

It  follows  that  ^  =  lOu^  X  -3=-30(l-  3x)^. 

4.  Prove  that  if  y  =  {ax  +  by,  and  u  is  a  positive  integer, 

5.  If  y={x+W2x+l)^  find  ^. 

We  have,  by  the  rule  for  differentiating  a  product, 

Using  the  results  of  Exs.  1  and  2  above, 

^=(a;  +  l)M2(2a;+l)5  +  (2x+l)6.4(a;+l)3 

=  4(.r  + l)3(2a;+ l)5{3(a:  + 1)  +  (2x+ 1)} 
=  4{x+lf{2x  +  l)H5x  +  4). 

^  a    Tf,,-i^±lll     dy__    {x  +  2){x+ir 
,    o.    -11  y-(2a;+l)6'   dx~     *      (2a;+l)'     ' 


7.   If  3^  =  (ax2  +  26x  +  cpo,  ^=20(ax  +  6)(aa:2  +  2ftx  +  c)9. 


GENERAL  THEOEEMS  ON  DIFFERENTIATION     39 

8.    Fill  up  the  blank  form  in  the  following  table  : 

f 


/(^). 

f'k^\ 

(l+x)^ 

u^ ;  ■  ■'   ' 

{i-xT- 

■> 

(l+a;Y 

{1-xr 

r-       ■•               l-^X 

[a  +  bxf 

u     ^-^'^^ 

{a  -  hxf 

'  (-?J 

(l+2a;2)4 

/ 


§  21.  The  Differentiation  of  x^,  when  n  is  any  positive  or 
negative  number. 

We  have  shown  in  §  19  that  when  n  is  Si  positive  integer,  the 
differential  coefficient  of  a;"  is  nx''~^. 

Also,  we  have  seen  in  §  20  that  this  result  is  true,  when  n  is 
any  negative  integer. 

We  shall  now  prove  that  it  is  true  in  general :  in  other  words, 
that  when  n  is  any  positive  or  negative  number,  the  differential 
coefficient  of  x^  is  nx^-^. 

Case  (i.).     Let  ?i  =  -,  a  positive  fraction  in  its  lowest  terms, 

J)  and  q  being  positive  integers. 

Then  x^  stands  for  the  real  positive  q^^  root  of  x^. 

Let 


p. 
■■x\ 


Then  we  have  y^^x^. 


The  differential  coefficients  of  both  sides  with  regard  to  x  must 
be  equal. 


Thus 

But 

Therefore 
Therefore  we  have 


dx 
d_ 
dx 
d_ 

dx 


--paf-\ 


dx 

lfj£     by§20,Prop.yL 


f  =  qf-' 


dx 


j-idy 
dx 


9.f~^'"£=P^'"'' 


40    DIFFEEENTIATION  OF  ALGEBRAIC  FUNCTIONS 
Therefore 


dy 
dx' 

If-' 

p    x'-^ 

'Hx^y-' 

= 

= 

=^^-- 

p 

y-- 

=  X\ 

dy 
Tx'- 

Thus,  when 


Therefore  our  formula  holds  when  n  is  any  positive  fraction. 
We  have  already  proved  that  it  holds  for  n  a  positive  integer. 
Therefore  it  is  true  for  any  positive  number. 
Case  (ii.).     There  remains  the  ease  of  a  negative  index,  integral 
or  fractional. 

Let  y  =  03"'",    where    m  >  0. 

Then  we  have        y  =  u''\      where    u  =  -- 


But 


dy  _  dy     du 
dx     du     dx 


Also  -^  =  mw"'~\    by  Case  (i.). 

.     ,  du         1 

And  -7-  =  — ^j 

dx         x^ 

by  the  rule  for  differentiating  a  quotient.     [Cf.  p.  36.] 

It  follows  that      -r-  =  mx~'^^'^  x  — 5 
dx  x^ 

Therefore  the  formula  holds  for  any  negative  number. 
Combining    these    results,    our    theorem    may   be   stated   as 
follows  :-^ 

If  y  =  xn      g  =  nx^-i, 

when  n  is  any  positive  or  negative  number.^ 

*  It  will  be  noticed  that  we  have  asaumed  that  w  is  a  rational  number.     The 
theorem  is  true  also  for  irrational  numbers. 


GENERAL  THEOREMS  ON  DIFFERENTIATION     41 

Ex.  1.    Fill  up  the  blank  column  in  the  following  table  :— 


r^> 


/(«;). 

f'{x). 

sjx 

\  ''■■'    ■ 

1 

Jx 

%> 

1 

x^ 

xi 

1 

^ 

1 

X2 

.  :2  v 

^          1 

x^ 

^'^-- 

1 

^  eyr 

1 

x' 

-  J  ^ 

1 

x^ 

-9f^ 

-  i 


^^ 


/4/i^ 


2.    If  .v= 


1-x 


(^a;     {\-xf 


We  have  y—-,  where  M=:l-a7.     Thus  y  =  u~'^,  where  M=l-a^. 


But 

It  follows  that 


dy_dy    du 
dx~  du    dx 

dx 
_J_ 


-^ jTT.,  find  -^. 


(1-x)^ 


3.   If  y  = 

We  have  y  —  ir^,  where  2t  =  3x  +  4 


It  follows  that 


dx 


-4x3 


~     (3a; +  4)4* 
4.   If  3^=  (ax +  6)",  and  ?i  is  any  positive  or  negative  number,  prove  that 


dy 
dx' 


■■naiax  +  hY 


42     DIFFERENTIATION  OF  ALGEBEAIC  FUNCTIONS 


5.   It  y=~ f-r>,  find  ^. 

We  shall  work  this  example  by  the  product  rule  instead  of  the  quotient 
rule.     This  method  often  saves  dividing  out  by  some  factors. 
By  the  product  rule  [cf.  §20,  Prop.  IV.], 

dx~^    ^   '  dx  {3x+4)^^ {3x  +  4f  dx^^^ ^'  ' 


Now  we  have  seen  in  Ex.  2  that    ^ 
d         1 


b 


dx  (3 

x  +  ^f        (3a; +  4)'*' 

And  it  is  easy 

to  show  that 

r+l)''*  =  2(a;  +  l). 

It  follows  that 

dy__ 
dx 

{x+\f      2{x+\) 
(3a: +  4)^'  (3a;  +  4)» 

(a;+l)(3a;+l) 

(3a; +  4)'*      ' 

6.    Fill  up  the 

blank  column 

in  the  following  table  :— 

/(^). 


1  + 


1  + 


(2a; +  3)* 

1 
(2a: -3)-* 

1 

(l-a;2)2 

1 

(1+X2)2 

1 


(l  +  a:)  '  2(l+a:)2 


(l-x)     2(l-a;)2 
1_ 

1 


sja^  +  x^ 

\la^  -  x^ 


fix). 


GENERAL  THEOREMS  ON  DIFFERENTIATION      43 
EXAMPLES  ON  CHAPTER  III 
1.    Find  ^  in  the  following  cases  : — 

(viii.)  y  =  {\  +  x^T. 
(ix.)  y  =  s/x'^  +  a^  +  \/x^  -  a^. 

(iv.)  y  =  (x  +  a)*.(a=  +  6)«.  (X.)  j,  =  _i=  +  -^^. 

X^ 


(i.) 

y-- 

y-- 
y-- 

M- 

-LV. 

'sIxJ 

(ii.) 

^sl2ax 

~x\ 

(iii.) 

=sl{x+\){x  +  2). 

(v.)  y=^\Yz-^  ^"^^'^  y- 


sl{i-xy 


,   ..  ,  ll+x  +  x^ 


...  {a-x)» 

2.  Find  the  gradient  at  the  point  {x^,  y^)  in  the  following  curves  : — 

(i.)  y'^  =  '^ax. 

(ii.)  a;2 +  2/2  —  ^2. 

(iii.)  -!±^2=1. 
(iv.)       2xy  =  c'^. 

3.  Prove  that  the  equations  of  the  tangents  at  (o^q,  ^q)  to  these  curves 

are  respectively 

(i.)  yyQ  =  2a{x  +  x^). 

(ii.)  xxQ  +  yyQ  =  a\ 

(iv.)  xyQ  +  yxQ  =  c\ 

4.  A  boy  is  running  on  a  horizontal  plane  in  a  straight  line  towards  the 
base  of  a  tower  50  yards  high.  How  fast  is  he  approaching  the  top,  when 
he  is  500  yards  from  the  foot,  and  he  is  running  at  8  miles  per  hour? 

5.  A  light  is  4  yards  above  and  directly  over  a  straight  horizontal  path 
on  which  a  man  six  feet  high  is  walking,  at  a  speed  of  4  miles  per  hour, 
away  from  the  light. 

Find     (i. )  The  velocity  of  the  end  of  his  shadow  ; 

(ii.)  The  rate  at  which  his  shadow  is  increasing  in  length. 

6.  A  man  standing  on  a  wharf  is  drawing  in  the  painter  of  a  boat  at 
the  rate  of  4  feet  per  second.  If  his  hands  are  6  feet  above  the  bow  of  the 
boat,  prove  that  the  boat  is  moving  at  the  rate  of  5  feet  per  second,  when 
it  is  8  feet  from  the  wharf. 

7.  A  vessel  is  anchored  in  10  fathoms  of  water,  and  the  cable  passes 
over  a  sheave  in  the  bowsprit,  which  is  12  feet  above  the  water.  If  the 
cable  is  hauled  in  at  the  rate  of  1  foot  per  second,  prove  that  the  vessel  is 
moving  through  the  water  at  a  rate  of  IJ  feet  per  second,  when  there  are 
20  fathoms  of  cable  out. 


44    DIFFEEENTIATION  OF  ALGEBRAIC  FUNCTIONS 

8.  If  a  volume  v  of  a  gas,  contained  in  a  vessel  under  pressure  p,  is 
compressed  or  expanded  without  loss  of  heat,  the  law  connecting  the 
pressure  and  volume  is  given  by  the  formula 

^yV  — constant, 
where  7  is  a  constant. 

Find  the  rate  at  which  the  pressure  changes  with  the  volume. 

dv         c^ 

9.  In  Boyle's  Law,  where  pv=.c^,  show  that  -t-=  — Ty     What  does  the 

negative  sign  in  this  expression  mean  ?  P        P 

10.  In  van  der  Waals's  equation 

( ^  + -^  j  (v  -  6)  =  constant. 

Prbvethat  *'-  "'-^' 


dp         f       a     2ah\ 


CHAPTEK   IV 

THE   DIFFERENTIATION    OF  THE   TRIGONOMETRIC   FUNCTIONS 

{The  angles  are  supj^osed  to  be  measured  in  Radians) 

§  22.     The  Differential  Coefficient  of  the  Sine. 
We  begin  with  the  value  x,  and  we  put 

y  =  sin  X. 

Then  we  take  8x  as  the  increment  of  x,  and  write  8y  for  the 

corresponding  increment  of  y. 

It  follows  that  y  +  8y  =  sin  {x  +  8x). 

8y  =  sin  {x  +  8x)  -  sin  x 

(       8x\   .    8x 
=  2  cos  (  a;  +  -^  j  sni  — . 

8y_  -        -""'  ■  ^ 

8x 


Therefore 


Therefore 


cos    X  + 


8x\ 
2; 


.    /8x\ 


8x 
T 


Proceeding  to  the  limit,  and  remembering  that 
Lt  (—pr-]  =  Ij  it  follows  that 


If  y  =  sin  X, 


dy 
dx 


=  cos  X. 


N.B. — When  y  =  sin  {mx  +  n), 


dy     dy  du       , 

-^  z=-f-  -—    where  u  =  mx  +  n, 

dx     du  dx 

_^/(sin  u)  du 

~      du      dx 

=  cos  w  X  m 

=  m  cos  {mx  +  n). 


46  THE  DIFFEEENTIATION  OF  THE 

Ex.  1.    Fill  up  the  blanks  in  the  following  table  : — 


Ax) 

sin  2x 

2sin| 

-  sin  3a; 
3 

^  sin  (4a; +  5) 
4 

3  sin  X  -  sin  3a; 

sin(l  -  x) 

f'{x) 

y^^l^ 

^ 

c^'^ 

l^p^  -3/**3|j 

.60^^'" 

2.    Prove  from  the  definition  of  -^,  that  when  y  =  sin(ma;  +  7i), 

dx 


dy 

dx' 


■mcoB{mx  +  n). 


%  23.   The  Diflferential  Coefficient  of  the  Cosine. 
We  begin  with  the  value  x,  and  we  put 

y  =  COS  X. 

Then  we  take  ^x  as  the  increment  of  x,  and  we  write  8?/  for  the 
corresponding  increment  of  y. 

It  follows  that  y  +  8y  =  cos  {x  +  8x). 

8y  =  cos  {x  +  8x)  -  cos  x 


Therefore 


2  sin 


in(^a;  +  ^j 


8x\   .    8x 
sm^. 


Thus 


8?/  .    /       hx 


/  .    8x' 
/  sin  — 


sm 


\  T 


Proceeding  to  the  limit,  it  follows  that 

If  y  =  cos X,  ~=  -  sin x. 

ax 

N.  B. — When  y  =  cos  {mx  +  n),    -f-=  -  '^n  sin  (mx  +  n). 
Ex.  1.    Fill  up  the  blanks  in  the  following  table  :— 


/(^) 

COS  2a; 

-2cos| 

1  cos  3a; 
3 

1  -  cos  2a; 

cos  3a; +  3  cos  a; 

icos(l-2a;> 

fix) 

2.    Prove  from  the  definition  of  -^,  that  when  y  =  cos(?na;  +  w), 

dx 


dy 
dx 


=  -msin(ma;  +  M). 


TKIGONOMETEIC  FUNCTIONS 
§  24.   The  Differential  Coefacient  of  the  Tangent. 
Let 


47 


sin  a: 

^  cos:r 


Then 


dx 


cl    .  .       d 

cos  ic-y- since  -  sin  a^ -7- cos  a^ 

dx dx 

cos'^a- 


cos^a^  +  sin^a; 
"~       cos%; 

~  cos% 

=  sec^a^. 

dy 
Thus,  if  y  =  tan  x,  ^  =  sec^x. 

N.B. — AVhen  ij  =  tan  {mx  +  n),    -j-  =  m  sec^  {mx  +  n). 
Ex.  1.    Fill  up  the  blanks  in  the  following  table  : — 


f{x) 

2tan| 

tan  \/x 

tan(a;2) 

1  +  s  tan  3x 

tan  2(1 -X) 

fix) 

2.    Prove  from  the  definition  of  -^,  that  when  y  =  tan  {mx +  11), 

dx 


dy 
dx 


■  msec^imx  +  n). 


J  7 

5-cotx=  -cosec^x;       -y- cot  (mx  +  n)  =  -mcoseG^(mx  +  n). 
dx  '      dx       ^  ^  ^  ^ 


From  these  three  results  it  is  easy  to  deduce  the  following  :- 

7?i  cosec^  (wa; 

d        ,  .         %\x\{mx-\-n) 

-Y-  sec  [mx  +  n)  =  m  — ~ r- 

dx         •  ^        cos'^{mx  +  n) 

d  ,  .  cos  (7nx  +  n) 

J- cosec  (mx  +  n)  =  -m^-~ .. 

dx  ^  '  sin^(m.T  +  ?i) 


d  smx 

-=-  sec  X  =  — ^ ; 
dx  cos^x 

d  cosx 

^-cosecx=  --7-7^ 
dx  sin^x 


§25.   Geometrical  Proofs  of  these  Theorems. 

All  these  cases  of  differentiation  may  be  discussed  geometri- 
cally. We  take  the  case  of  the  tangent.  The  reader  is  recom- 
mended to  work  out  for  himself  the  cases  of  the  sine  and  cosine. 


48 


THE  DIFFERENTIATION   OF  THE 


Let  z_MOP  be  the  angle  0  radians,  and  let  OM  be  1  unit  in 
length. 

Let  ^POQ  be  W^  and  let  QPM  be  perpendicular  to  the  line 
OM  from  which  6  is  measured. 
Let  PN  be  perpendicular  to  OQ. 
Then 

8(tan6')  =  PQ 

=  PNsec:LNPQ 


Fig.  10. 


Thus 

S(tan^) 


he 


and  proceeding  to  the  limit, 

d_ 

cie 


=  PNsec(^  +  56>) 

=  OPsec(6'  +  S6')sinS(9 

=  sec  ^  sec  (6' +  8^)  sin  8(9. 

=  sec^sec(^  +  8^)(-'||^), 


tan  6  =  sec-^. 


Examples.     Find  ^  in  the  following  cases  : — 
dx 

(i.)  y  =  '2a&\n{hx-\-c)Bin[hx -c). 

(ii.)  y  =  a;- cos  2a;. 

(iii.)  ?/ =  tan  3a;  +  cot  3a;. 

, .     ,         sin  2a;  -  sin  x 

(iv.)  v  = . 

^      '  ^  cos  a; 

(v.)  y  =  x'^mxi'^x. 

(vi.)  y  =  x'^B\nnx. 

(vii.)  y  =  sin-*"  a;  cos^  a;. 

( viii. )  y  =  sec"^(ax  +  6)  +  cosec2(cx  +  d). 

§  26.   The  Graphs  of  the  Trigometrical  Functions. 

The  results,  which  we  can  deduce  from  the  differential  co- 
efficients of  the  functions 

sinic,    cos  a;   and   tana:, 
should  be  compared  with  the  information  to  be  obtained  from 
the  graphs  of  these  functions. 

These  graphs   are   given — when    the   angle    is    measured    in 
degrees  and  with  a  suitable  scale — in  Figs.  11,  12  and  13. 

It  must  he  noticed  that  when  x  is  the  number  of  degrees  in  the  angle 

whose  sine  is  y,  the  differential  coefficient  -^  is  not  cos  x. 


TRIGONOMETRIC  FUNCTIONS 


49 


It  is  a  good  exercise  for  the  reader  to  show  that  in  this  case 


dx     180 


cosx. 


A  similar  change  has  to  be  made  in  the  differential  coefficients 
of  the  other  Trigonometrical  Functions,  when  they  are  not 
measured  in  radians. 


^  "1  t/f  pcis-x 


'^ 


t 


h 


y  =  co%x. 
Fig.  12. 


However,  the  general  behaviour  of  the  functions — when  they 
are  increasing,  and  when  decreasing;    when    they  reach   their 
C.C.  D 


50 


THE  DIFFERENTIATION   OF  THE 


maxima  or  minima,  if  such  exist ;  when  their  graphs  are  con- 
cave upwards,  or  convex  upwards,  etc. — can  be  seen  from  these 
figures. 


Ex.    In  the  Four-Figure  Tables,  we  are  told  that 
sin  46°     =-7193 
and        sin46°6'=-7206. 

Compare  this  result  with  that  obtained  by  the  Calculus  method. 
Use  cos  46°  =-6947. 


INVERSE  TRIGONOMETRICAL  FUNCTIONS        51 


THE   INVERSE  TRIGONOMETRICAL  FUNCTIONS 

§  27.    The  Diflferentiation  of  the  Inverse  Sine. 

To  any  value  of  x,  lying  between  -  1  and  +  1,  there  corre- 
sponds an  infinite  number  of  angles  which  have  this  value  x  for 
their  sine.  If  y  is  the  circular  measure  of  one  of  these  angles, 
then  sin?/  =  a: 

is  the  equation  connecting  x  and  y. 

If  we  give   to  //  a  number  of 
values,   we   can   obtain   from   the 
Tables  the  corresponding  values  of 
X,  and  in  this  M^ay  plot  the  curve 
sin  y  =  x. 

It  is  clear  that  it  is  a  periodic 
curve  of  period  2xr  in  y,  and  that 
it  could  be  derived  from  the  sine 
curve  by  placing  this  curve  along 
the  axis  of  y,  instead  of  along  the 
axis  of  X. 

Another  way  of  drawing  the 
curve — and  this  is  common  to  all 
such  inverse  curves— is  to  fold  the 
paper,  on  which  the  curve 

y  =  sin  X 
is  drawn  about  the  line 

y  =  x, 
and  this  sine  curve  will  then  co- 
iucide  with  the  curve 
sin  y  =  X. 

It  is  convenient  to  have  a  name  fig,  u. 

and  a  symbol  for  this  functional 

relation.     If  y  is  the  circular  measure  of  the  angle  whose  sine 
is  X,  y  is  said  to  be  the  inverse  sine  of  x,  and  the  notation  adopted 


y 

=  G/ 

n 

'x 

y 

IT- 

\\ 

\ 

\ 

s 

\ 

\ 

\ 

TT. 

, 

J 

/ 

/ 

/ 

/ 

• 

/ 

-1 

/ 

0 

1 

X 

/ 

/ 

/ 

- 

2 

\, 

\ 

\, 

\ 

N 

- 

-ttH 

IS 


Part  of  the  curve 
is  given  in  Fig.  14. 


y  =  sin  ^x. 
y  =  s\n~'^x 


52  THE  DIFFERENTIATION   OF  THE 

To  save  ambiguity  and  to  make  the  function  single-valued — 
that  is  to  give  only  one  value  of  y  for  one  value  of  x — it  is  an 
advantage  to  restrict  the  symbol 

sin~^a; 

to  the  number  of  radians  in  the  angle  between  -  ^  and  -^  whose  sine 
is  X. 

With  this  notation  the  curve 

y  =  sm~^x 
would  be  the  part  of  the  curve  on  Fig.  14  which  lies  between 
the   values   --  and  -  of  y.     This  is  drawn  on  Fig.   14  in  a 

heavier  line. 

We  shall  use  the  symbol      sin'^a; 
for  this  value  only :    that  is,  the  angle  whose  sine  is  x  is  to  be 

measured  in  radians  and  to  lie  between  -  ^  and 


»   We  proceed  to  the  differentiation  of  sin^^x. 

We  begin  with  the 

equation 

^  =  sin-ia^.     (^-|<y<|^ 

Then 

sin  y^x. 

On  differentiating  1 

both  sides  of  this  equation  with  regard  to  ic, 

we  have 

d    .           d       , 
dx       ^     dx 

But 

d    .          d    .         dy 

=^^^4'- 

It  follows  that 

dy    , 

and 

dy    ■    \ 
dx    cos  y 

{ 


TT  TT 

But  we  know  that  sin y  =  x^  and  that  -  -^<y  <Tt' 


INVERSE  TRIGONOMETRICAL  FUNCTIONS        53 

Therefore  we  must  have 


cos  y  =  Vl  -  «^ 
the  square  root  being  taken  with  the  positive  sign. 

Hence  ^ 


1 


Therefore  the  differential  coefB.cient  of  sin  ^x  is 


Vl-x2 


It  will  be  noticed  that  if  we  take  the  complete  curve  for  the 

inverse  sine,  instead  of  the   portion  from  --  to  ^  only?  the 

gradients   at   the    points   where    x  =  const,   cuts   the    curve    are 
alternately  I 


sl\-X^ 


Ex.  1.   If  y=sin 

We  have 

But 

It  follows  that 


-1?  <^y^     1 

a    dx    ^a2-x2' 


=:sm-^ 

u, 

where    u  - 

X 

a 

dy 
dx~ 

dy 
^d^i 

du 
""dx- 

dy 
d~x' 

^VT 

1          1 

rrrrmrX- 

1 

x/a2-x2 

2.   If  ,  =  si„-.(.-'),   1  =  ^- 

We  have  y  =  s,\n-'^u,     where    u  =  x^, 

dy  _dy^    du 
^^^  di-du'^dx 

It  follows  that  |  =  -^^x2. 

'  2x 


s/T 


If  v  =  sin   ^    — =r- 


V  \/2  /     c?^        s/l+2x-Q^ 


54  THE  DIFFERENTIATION  OF  THE 

4.    Fill  up  the  blanks  in  the  following  table  : 


/(^). 

/'(^). 

sin-Ml+^) 

8in-i>yic 

^■"-'^ 

^'"-W^ 

--(-) 

§  28.   The  Differentiation  of  the  Inverse  Cosine. 

To  any  value  of  x,  lying  between  - 1  and  + 1,  there  corre- 
sponds an  infinite  number  of  angles  which  have  this  value  x  for 
their  cosine.     If  y  is  the  circular  measure  of  one  of  these  angles, 

GOsy  =  x 

is  the  equation  connecting  x  and  y. 

This  relation  is  also  expressed  by  the  notation 

y  =  cos~^x, 

and  y  is  said  to  be  the  inverse  cosine  of  x. 


Part  of  the  curve 


=  cos  ^x 


is  given  in  Fig.  15. 

In  the  case  of  the  inverse  cosine  it  is  again  convenient  to 
make  the  function  single-valued.     For  this  purpose  it  is  best  to 

restrict  the  symbol  , 

•^  cos  ^x 

to  the  number  of  radians  in  the  angle  between  0  and  tt  whose  cosine  is  x. 
With  this  notation  the  curve 

y  =  cos~'^x 

would  be  that  part  of  the  curve  in  Fig.  15,  lying  between  the 
values  0  and  tt  of  y.  It  is  drawn  with  a  heavier  line  in  that 
figure. 


INVERSE  TRIGONOMETRICAL  FUNCTIONS        55 


We  proceed  to  the  differentiation  of  the  inverse  cosine. 
We  begin  with  the  equation 

y  =  cos~i  X.     (0  <  y  <  tt) 
Then  cos?/  =  a:. 

On  differentiating  both  sides  of  this  equation  with  regard  to 


X,  we  have 


-7-  cos  ?/  =  ^-  a;  =  1 


dx 


dx' 


But    ,-  cos  ?/  =  -7-  cos  yx-r- 
dx  cii/  dx 


It  follows  that 


sm^ 


and 


dx 

dy 

dx 


1. 


1 

sin 


But  we  know  that  cos  y  —  x 
and  that  0  <  ^  <  tt. 

Therefore  we  must  have 


sin?/  =  \/l  -x^, 

the  square  root  being  taken 

with  the  positive  sign. 
Hence 

dji^  __   1 

dx~     Vf^^* 

Therefore    the   differential 
coeflBlcient  of  cos~ix  is 


y=cos 

-'x 

J' 

•TT 

s. 

N 

\ 

\ 

\ 

\ 

\ 

\ 

-1 

0 

1 

X 

/ 

/ 

K 

' 

/ 

La 

2 

/ 

/ 

—71 

sJl-X^ 


Fig.  15. 


This  result  could  have  been  deduced  from  §  27,  since,  with  the 
notation  we  have  adopted, 


-U'== 


56 


THE  DIFFEEENTIATION  OF  THE 


For  example, 


sin" 


cos" 


l^TT    1 

2~6 
2~3' 


x/2. 


COS" 


It  will  be  noticed  that  if  we  take  the  complete  curve  for  the 
inverse  cosine,  instead  of  only  the  portion  from  0  to  tt,  the 
gradients  at  the  points  where  x  =  const,  cuts  the  curve  are 
alternately  1 


Vi-: 


Ex.  1.    If  y  =  cos- 


,/x\     dy. 


ay     dx 


Va^  -  x'' 


2.    If  y  =  cos-^x^),   ^  = 


3:r2 


3.    If  y  =  cos- 


l  +  x'J'   dx     1+x^' 


1  -X- 
In  this  example,  we  have  y  =  cos~^u,  where  ?t=         ^. 

But 


rfy  _  dy    dii 
dx  ~  du    dx 


Also 
And 

It  follows  that 


dy^ L_. 

du        sfl^^^ 


2x 


du_{l+x'^){- 2x)  -  (I  - a:^)2x 

dx~  (1+X2)2 

4a; 


(1+0:2)2 
dy        2 


dx     l+x"^' 
4.    Fill  up  the  blanks  in  the  following  table 


Ax) 


fix) 


C08-^(l  -X) 


■'Vi 


cos~^{2a;-  1) 


INVERSE  TRIGONOMETRICAL  FUNCTIONS        57 

§  29.    The  Differentiation  of  the  Inverse  Tangent. 

To  any  value  of  x  lying  between  -  oo  and  oo  ,  there  corresponds 
an  infinite  number  of  angles  which  have  this  value  x  for  their 
tangent.     If  y  is  the  circular  measure  of  one  of  these  angles, 

tan  y  =  x 

is  the  equation  connecting  x  and  y. 

This  relation  is  also  expressed  by  the  notation 

«/  =  tan~^ic, 

and  y  is  said  to  be  the  inverse  tangent  of  x. 

Part  of  the  curve  y  =  t3inr'^x 

is  given  in  Fig.  16. 

y=tan-\ 


X             X               1^ 

IT 

^ ^_l 

=-======^============^========±^=^= 

^ " 

^--'^ 

^^ 

^ 

^ 

-4            -3            -2             H              /q              1                2              3              4          X 

.                  ^^ 

h                                  ^^                               +          + 

— f— ■^'' 

1  1          iT    H                          M 

±  -             ^=1  . 

Fig.  16. 


In  the  case  of  the  inverse  tangent  it  is  again  convenient  to 
make  the  function  single-valued,  and  this  is  done  by  restricting 
the  symbol  tan"^^;  to  the  number  of  radians  in  the  angle  between 


and  -  whose 


IS  X 


With  this  notation  the  curve 

y  =  tan'^a; 
would  be  the  part  of  the  complete  curve  of  the  inverse  tangent 
for  the  values  -  -  to  -  of  ?/.     It  is  this  part  of  the  curve  which 

is  given  in  Fig.  15. 

We  proceed  to  the  differentiation  of  the  inverse  tangent. 


58  THE  DIFFERENTIATION   OF  THE 

We  begin  with  the  equation 

?/  =  tan  ^x.     \-T2^<y<2) 
Then  tan  ij  =  x. 

On  differentiating  both  sides  of  this  equation  with  regard  to  x, 

we  have  ^  ^/ 

-r-tanv=  -v-a;=  1. 
ax        ^     ax 

-D  ,  d  ^  cl  dy  „  (Zy 

rJut  -7-tan?/  =  -^tanwx -y^  =  sec-2/^. 

dx        '       dy        ^     dx  ^  dx 

And  sec^^  =  1  +  tan-y  =  1  +  a:^. 

Therefore  -f^  =  ^, —  „. 

dx     l+x^ 

Therefore  the  differential  coefficient  of  tan~^x  is  — —  „. 

1  +x2 

It  will  be  noticed  that  if  we  take  the  complete  curve  for  the 

inverse  tangent,  instead  of  only  the  portion  between  -  ^  and  -, 

the  gradients  at  the  points  where  x  =  const,  cuts  the  curve  have 
1 


the  same  value 


1  +X-' 


Ex.1.   Ify=tan-i-,  ^  =  -0^- 


2.    If  y  = 

tan" 

-0 

dy            1 
'   dx        1+x^' 

We  have 

y  =  tan-i  ^l, 

But 

dy    dy    du 
dx~du    dx' 

Also  !^=  -L  -_^_,. 

du     l  +  u^     1  +  x^ 

A    J  du         1 

And  =_ 

dx        x^ 

It  follows  that  ^=  -  — ^-,. 

dx        l  +  x^ 

It  will  be  noticed  that  the  angle  whose  cotangent  is  x  has  -  for  its 

X 

tangent.     This  example  shows  us  that  the  differential  coefficient  of  the 

inverse  cotangent  of  x  is -> 

l  +  x^ 

This  result  would  also  follow  from  the  equation 
tan~i  a;  +  cot"^  x  =  ^t 


INVERSE  TEIGONOMETEICAL  FUNCTIONS        59 

which  is  true  if  we  take  cot~^a:  to  lie  between  0  and  tt,  wliile  tan~^x  lies 
between  -  ^  and  ^. 

3.    If  y  =  Un-^x'),  ^  =  ^,' 
^  ^     "  dx     1  +  x^ 

4    If     =  — tan-i?:^,   ^y^        ^ 

^    \/S  ^"        \/3   '   c^^    x'^  +  x+V 

In  this  example,  we  have 

2.-1  1,  2a;+l 

V  =  -7=tan^?t,     where     ?t  =  — ,-  • 


But 


(Zy  _  dy    du 


Working  out  ^,  we  find  that  it  is  equal  to 


n/3 


du 
And 

It  follows  that 


du_2 
dx~  ^^ 
dy_        1 


2(a;2  +  a;+l) 


dx    x^  +  x+\ 


5.    If  y  =  xtdM~^x,  -^t=~— — ^  +  tan   '^x. 

Using  the  rule  for  differentiating  the  prodvict  of  two  functions,  we  have 

dy         d  ,        ^        .        .      d 
-^  =  a;-r- tan~^x  +  tan  ^x-r-x 
dx       dx  dx 

— .;  +  tan~^a;. 

\+x^ 


6.    Fill  up  the  blanks  in  the  following  table  : 

f{x) 

tan-M2a;-l) 

--(^) 

tan-iJ= 

sjx 

Itan-i^ 
a            a 

a;Han-ia:2 

fix) 

EXAMPLES  ON  CHAPTER  IV 

1.    Differentiate  the  following  functions  : — 

(i.)  sin^a:  +  cos^a:. 

(ii.)  tanx  +  Ktan^a;. 

(iii.)  sec-a^  +  tan^a;. 
(iv.)  cosee^ar  +  cot^ic. 

(v.)  -1+-^"!^. 
1  -  sin  X 

\-co^x^ 

1  +  cos  x 


60        INVERSE  TRIGONOMETRICAL  FUNCTIONS 

rt    T£  sin  a;  ^,    ^  dy     cos^a^-sin^a; 

2.  If  y=z — 7 ,  prove  that  -^  =  -, . ,„. 

1  +  tanx    ^  ax    (cos  a:  +  sin  a;)'^ 

3.  If  y  =  cos(a;^),  prove  that  --^=  -  Sa;^  sin  (a;^),  and  find  -~  when 

(i.)  y  =  a;'"sina;". 
(ii.)  y  =  a:"*  cos  a;", 
(iii)  2^  =  a;"*  tan  a;". 

4.  Differentiate  the  following  functions  : — 

(i.)  (a;2+l)  tan-^a^-a;. 
(ii.)  X sm~^ X  +  sfl^^^. 

(iii. )  tan-i  Z'^  +  ^V        (Put  six  =  tan  d,  si  a  =  tan  a. ) 

V  1  -  \lax  J 


(iv.)  tan 


\\-x+x^y 


(v. )  cot-i  (  ^+'^^+-^-  \ .        (Pixt  a;  =  tan d. ) 


5.  A  particle  P  is  revolving  with  constant  angular  velocity  w  in  a  circle 
of  radius  a.  The  line  PM  is  drawn  from  P  perpendicular  to  the  line  from 
the  centre  to  the  initial  position  of  the  particle.  Find  the  velocity  and 
acceleration  of  M. 

6.  If  the  position  of  a  point  is  given  at  time  t  by  the  equations 

x  =  a{u}t-\-?,\n(j3t), 

y  =  a{\  -cosw<), 
where  a  and  w  are  constants,  find  its  component  velocities  and  accelerations, 
and  its  direction  of  motion  at  the  time  t. 

7.  Prove  that  when 

dx^  xs!x^-\ 

and  that  when  a;>  1,     -5-  (sec~^a;)  = 


dx  xsl:x?-\ 

and  illustrate  your  results  from  the  graph  of  the  inverse  secant. 

8.    Prove  that  when 

^  A       d  ,  ,    ,  1 

a;  >  1 ,     -J-  (cosec~^  x)= 7— — » 

'     dx^  xslx^-l 

and  that  when  x<-\,     ^-  (cosec-^a;) 


ax  X  V  a:^  -  ] 

and  illustrate  your  results  from  the  graph  of  the  inverse  cosecant. 


CHAPTER  V 

THE  EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS — MAXIMA  AND 
MINIMA — PARTIAL  DIFFERENTIATION 

§  30.   Introductory. 

There  remain  two  important  functions  which  we  must  learn 
to  differentiate  :  the  logarithm  of  x  to  the  base  a,  and  its  inverse 
function  a"". 

We  shall  find  that  there  is  a  particular  number  denoted  by  e 
for  which  the  Logarithmic  Function 

log,.?; 
and  the  Exponential  Function 

are  of  the  greatest  importance. 

The  differential  coefficients  which  we  require  can  be  obtained  much 
more  quickly  with  the  a^id  of  Infinite  Series,  and  those  who  are  familiar 
with  that  branch  of  Algebra  will  probably  prefer  the  usual  method  of 
finding  them  given  in  the  Appendix.  In  the  articles  which  follow  we 
obtain  them  without  using  more  than  Elementary  Algebra  and  the 
Logarithm  Tables.  It  is  true  that  this  discussion,  in  one  or  two  points, 
is  not  quite  rigorous.  Still  those  for  whom  the  rigorous  treatment  is 
suitable  will  get  it  in  their  later  course.  Those  who  do  not  carry  their 
study  of  the  Calculus  further  will  yet  have  obtained  a  working  knowledge 
of  the  meaning  of  the  new  functions  and  a  complete  enough  grasp  of  the 
application  of  the  Calculus  to  them. 

The  following  formulae  in  logarithms  are  supposed  known : 

logxMN  =  logM4-logN,  (1) 

log  M.  log  M- log  N,  (2) 

logM''  =  rtlogM.  (3) 

These  are  true  for  any  base.     All  the  numbers  are  supposed  to 
be  positive. 


62    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS 

By  a  simple  application  of  the  Index  Laws,  another  formula 
is  obtained,  which  allows  us  to  change  logarithms  from  one  base 
to  another.     This  formula  is 

log^N^l^;  '  (4) 

If  we  put  N  =  Hn  this  equation,  we  have 

\og,bx\og,a^l.  (5) 

Thus  we  can  write  (4)  in  the  form 

log,N  =  log„/>xlog,N.  (6) 

Since  log^a;  =  log^h  \og,x, 

we  have  —  \og„x  =  log^fi  -r.  logip'. 

It  follows  that,  if  we  know  the  differential  coefficient  of  the 
logarithm  of  x  to  any  base  a,  we  can  write  down  that  for  any 
other  base  b,  it  being  of  course  understood  that  the  bases  a  and  b 
are  independent  of  x. 

In  the  discussion  which  follows  we  shall  first  find 
d 


dx 


logio^' ; 


but,  before  we  can  do  so,  it  Avill  be  necessary  to  learn  something 
about  the  behaviour  of  the  expression 

1 


^''logio(l+,^^ 


as  n  gets  larger  and  larger. 

In  the  work  on  Algebra,  which  we  are  omitting,  it  is  proved 

rigorously  that  the  number  (1  +  -)    continually  increases  as  7^ 


increases,  and  that  when  7i  ->  go  ,  it  has  a  definite  limiting  value. 
In  other  words,  Lt  ( 1  +  -  ) 

is  a  definite  number. 

It  is  true  that  this  number  is  incommensurable,  but  its  value 
can  be  obtained  to  as  close  a  degree  of  accuracy  as  is  required. 
Correct  to  7  places  of  decimals  it  is  27182818. 

This  number  is  denoted  by  e.  It  is  the  base  of  the  Napierian 
or  natural  system  of  logarithms. 

From  the  result  that  „ 

Lt  (l+-)  =e, 


PARTIAL  DIFFERENTIAL  COEFFICIENTS 


63 


Lt  n  log„(  1  +  - )  =  log„^, 


it  follows  that 


where  a  is  any  base. 

In  particular,  if  we  take  the  base  10  and  look  up  the  logarithm 
of  2-7182818  in  the  Tables,  we  find  that 

Lt  wlogio(l+^)  =  -4342945. 

Without  assuming  the  truth  of  any  of  the  above  work,  we 
shall  now  see  what  information  the  Tables  give  us  regarding  the 
expression  /       i\ 

§  31.   The  Expressions 

(l+^     and   nlogi,(^l  +  -). 
In  the  accompanying  tables  the  approximate  values  of 

«log,„(l+J,) 

are  given  for  7i=  1,  50,  100,  500,  etc.  The  figures  in  Column  II. 
are  calculated  from  7-Figure  Logarithm  Tables ;  those  in 
Column  III.  from  8-Figure  Tables. 

TABLE 


Showing  the  A'alue  of 

nlog,o(^l+^^. 

n 

7-Figure  Tables. 

8-Figure  Tables. 

1 

50 

100 

500 

1000 

2000 

3000 

4000 

5000 

6000 

7000 

8000 

9000 

10000 

•3010300 

•43001 

•43214 

•4338 

•4341 

•4  3  4  2 

•4341 

•4344 

•4345 

•4  3  3  8 

•4347 

•4344 

•4347 

•4  3  4 

•30103000 

•430009 

•432137 

•4  3  3  8  6  5 

•43408 

•43418 

•43425 

•43424 

•43425 

•43428 

•43428 

•43424 

•43425 

•4343 

64    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS 


It  will  be  seen  that  the  values  we  have  obtained  for 

«Iog,o(l+^) 

increase  with  n  until  we  reach  ?i  =  2000  in  the  second  column, 
and  n  =  3000  in  the  third. 

The  oscillation  that  we  meet  there,  and  in  some  of  the  later 
numbers,  is  due  to  the  fact  that  in  7-Figure  Logarithm  Tables 
the  seventh  decimal  place  is  only  the  nearest  value,  and  may  err 
to  the  extent  of  "5  either  way.  When  the  logarithm  is  multiplied 
by  1000,  the  unknown  error  in  the  product  comes  within  "5 
either  way  of  the  fourth  decimal  place.  In  the  products  from 
2000  to  9000  this  may  affect  the  fourth  decimal,  and  even  the 
third. 

The  same  argument  applies  to  the  results  in  Column  III. 
from  8-Figure  Tables,  and  in  this  way  the  oscillations,  when 
71  =  3000  and  4000,  and  when  71  =  7000  and  8000,  can  be 
explained. 

To  avoid  this  source  of  error,  and  to  show  still  more  clearly 
the  behaviour  of  the  expression 

»log.„(l+,') 

as  n  increases,  the  following  table  has  been  calculated,  cairect  to 
ten  places,  using  15-Figure  Logarithm  Tables. 

TABLE 

Showing  value  of  n  logio|  1  +  r  )  correct  to  10  places. 


n 

nlog,o(l+y. 

1 

10 

25 

50 

100 

500 

1000 

10000 

100000 

•3  010299957 
•4  139268516 
•4  258334825 
•4  300085881 
•4  321373783 
•4  338607656 
•4  340774793 
•4  342727686 
•4  34294  2  648 

For  ?i=  1,000,000,  we  find  7ilogj 


1+-    =-434294460. 


PARTIAL  DIFFERENTIAL  COEFFICIENTS         65 

It  will  be  seen  from  these  results  that  we  may  safely  assume 
that  as  n  gets  larger  and  larger 


log>o(l+i) 


gets  very  near  the  number  0-4343 ;  and  we  find  from  the  Tables 
^^^^  logio2-7l8  =  -4343. 

We  shall  therefore  assume  that 


.00    \ 


Lt  ( 1  +  -Y 


exists,  and  we  shall  denote  it  by  e. 

We  shall  take  as  our  approximation  to  e  the  number  2  71 8, 
and  we  shall  take  for  our  approximation  to  log^Qe  the  number 
•4343. 

We  are  now  able  to  proceed  to  the  differentiation  of  the 
logarithm  ^of  x  to  any  base.  We  shall  begin  with  the  base  10, 
and  then  find  the  differential  coefficients  of  log^a:  and  log„a:. 

From  these  results  we  shall  readily  obtain  the  differential 
coefficients  of  e""  and  a*. 

§32.    The  Differentiation  of  log^oX.     (Cf.  App.  p.  130.) 
We  begin  with  the  value  x,  and  we  put 

y  =  \og^ox. 

Then  we  take  the  increment  8x  of  x,  and  we  write  8y  for  the 
corresponding  increment  of  y. 

It  follows  that  y  +  8y  =  log^Q  (x  +  8x). 

Therefore  we  have 

^f/=^^og^o{^-i-8x)-\og^QX 

=  log,o(l+f). 
Therefore  |  =  ilog.(l4) 

X 

Now  put  7i  =  ^  on  the  right-hand  side, 
c.c.  E 


6Q    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS 

Remembering  that  x  is  fixed,  we  see  that  when  the  increment 
of  X,  namely  8x,  is  made  smaller  and  smaller,  n  gets  larger  and 
larger. 

Also  as  8x->0,  n^-  co  . 

Therefore  we  have 

Lt  f^V-xLt  [n\ogJ\  +  ^^ 
It  follows  from  §  31,  that 


dx 


iQgipg 


loff  e 
Thus  the  differential  coefficient  of  logi^x  is  _^io_. 

§  33.   The  Differentiation  of  logeX  and  logaX.    (Cf.  App.  p.  130.) 
Since  l«g«^  =  T^'     P-§30(4)] 

it  follows  that  ^^og,^  =  Io~.  T^  l^Sio^ 

log^o^""^   ^^i^"- 

Therefore  -^  lege  x  =  - . 

dx  X 

Again  we  have  log.^  =  }^.     [Cf.  §  30  (4)] 

It  follows  that        i^^^aX-^J^\og^x 

1        1 

log.a     x 

Therefore  ^  log.x  ^  ^^  =  ^^-f .     [Cf.§30(5)] 

In  Elementary  Trigonometry  it  is  convenient  to  write  logN 
for  log^oN.  In  the  Calculus  and  in  Higher  Mathematics  we 
usually  write  log  N  for  log.N  :  that  is,  we  only  insert  the  base 
of  the  logarithm  log„.T 

when  the  base  a  is  different  from  e. 

However  sometimes  we  shall  insert  the  base  e,  if  it  is  necessary 


PARTIAL  DIFFERENTIAL  COEFFICIENTS 


67 


to  emphasise  the  fact  that  logarithms  are  being  taken  to  that 
base.     With  this  notation  the  results  of  this  section  are  written 


logae 


dx^      ^  ^     xloga        X 


(x>0) 


(x>0) 


The  equation  -T-\ogx  =  -  is  of  the  greatest  possible  importance, 
ax  X 

Ex.  1.    If  y  =  los{ax  +  b),       -^^     ^  ,. 
^        ^^  "       dx    ax  +  b 


We  have 
But 

Therefore 


y  =  logii,     where     u  =  ax-\-b.    &1.    ,■ 
dy  _  dy    du 
dx    du    dx' 


dy     I  a 

-^—-xa  = 5-- 

dx    u  ax  +  b 


2.    If  .  =  I„g,a.=  .2...o,,   1  =  J^|±^. 


3.  If  y=;  log  sin  x, 

4.  If  y  =  log  cos  a-, 


dy 
dx 

dy 
dx 


=  Q,OXiX. 


tana;. 


5.    If  y  — log  tan'-, 

dy_    1 
dx    sin  x' 

We  have  y  -  log  li, 

where 

u  =  tan  - . 

It  follows  that 

dy    K,     1 

dx     u    2eos-2| 

1 

2  sin  1  cos  1 

1 

smar 

6.    Ify  =  lofff(x), 

dy     f'(x) 
dx     f'(xj" 

7.    Fill  up  the  blanks 

in  the  following  table 

Ax) 


fix) 


log(l-x) 


log(l  +  cc) 


^-(}^:) 


log  (a; -a) 


log(x  +  a) 


log 


x  +  a 


log 


68    EXPONENTIAL  AND  LOGAEITHMIG  FUNCTIONS 

We  have  y  =  log(a2-a:2) -log(62_3.2)^ 

Therefore  ^=  _     2^,4-    ^^ 


2x 


(a2-x2)(  62-^:2) 


9.  If  y=log(.:i^/5^#,,     1=^,- 

10.  Tf  yr^lng  J«-^>cosx     (/y^     a6sin:c 

§34.    The  Differentiation  of  e^.     (Cf.  App.  p.  130.) 

Let  y  =  e^ 

Then  we  have  log  y  =  x  log  e. 

Therefore  log  y  =  x,    since    log/  =  1 . 

Differentiating  both  sides  of  this  equation  with  regard  to  x, 


3  nave 

ix^'^^y-^- 

But 

d 
dx 

log  :y-|  logy  x| 

_1  dy 
~  y  dx' 

It  follows  that 

1^=1 

y  dx 

Therefore 

t-y-^- 

Thus  the  differential  coefficient  of  e^  is  e^. 

d_ 
dx 


The  equation  -j-e''  =  e^  is  of  the  greatest  possible  importance. 


Ex.   1.   If  y  =  e^^, 

g=.e-. 

We  have 

y= 

=  e",     where    u  =  mx. 

It  follows  that 

dx 

2.    If  y  =  e«^+^ 

dx 

3.    If  y  =  e'^smbx, 

-1-  =  e"^  [a  sin  hx  +  &  cos  6ic]. 

PARTIAL   DIFFERENTIAL   COEFFICIENTS  69 

4.    If  7/  =  e~'^''smbx,  -^  =  e~'^[bcosbx-asmbx]. 


5.    If  y  =  rte^ 

t-^y 

6.    If  y  =  e-\ 

t=^^y 

7.    If  y  =  xe^\ 

x|=(l+2.V. 

§  35.    The  Differentiation  of  a^. 

Let 

«/  =  «'. 

Also  let 

log^«  =  m. 

Then 

€"^  =  0. 

Therefore  we  have 

y  =  e'^\ 

It  follows  that 

dx 

Therefore 

^a^  =  allege  a. 

It  must  be  noticed  that  the  number  a  in  this  formula  is  a 
constant  and  independent  of  x. 

Ex.  Prove  that  the  differential  coefficient  of  a^  is  a^loga,  by  taking 
logarithms  of  both  sides  of  the  equation 

and  then  differentiating. 

§  36.   Logarithmic  Differentiation. 

We  have  already  obtained  a  general  rule  for  the  differentiation 
of  a  product  or  quotient.  We  are  now  able  to  prove  another 
method  which  often  leads  more  quickly  to  the  result.  This 
method  is  called  Logarithmic  Differentiation. 

Let  y  =  uvw. 

Then  log  y  =  log  u  +  log  v  +  log  w. 

■■■   |i;(log2/)  =  ^(log«)  +  i(log^)  +  |(log«'). 
d  ,.        .dy      d  ,,        .du      d  ,,       .  dv      d   ,,        .dw 

1  dy     \  du     1  dv     1  dw 
y  dx     u  dx     V  dx     m  dx ' 


Multiplying  up  by  y,  we  have  the  value  of  -/-. 


70    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS 

In  other  words,  before  differentiating  an  equation  involving  the 
product  or  quotient  or  powers  of  other  expressions,  take  logarithms  of 
both  sides  of  the  given  equation. 

Thus  when  we  are  given  an  equation  involving  the  product,  or 
quotient,  or  powers  of  several  expressions,  it  is  often  an  advan- 
tage to  take  logarithms  of  both  sides  of  the  given  equation 
before  differentiating. 


Ex.  1.    li  y-- 

^V[ 

-x^ 

+  X-2' 

dy                    2x 

dx        J{l-x'){l+x^f 

We  have 

\ogy-- 

=  ^log(l-x2)_liog(i  +  a.2), 

Therefore 

1  dy            X           X 
y  dx~     1-x^     l  +  x^ 
2x 
-      l-x^' 
dy                    2x 

dx        s/{\-x^){l+x-f 


2.    If  y  =  {l-x)Hl-2x)\  find  ^^. 

dx 


We  have  log  y  =  3  log  ( 1  -  ic)  +  4  log  ( 1  -  2x). 

\dy_  __3 8 

y  dx         \-  x     \-2x 

(11 -14a;) 


Therefore 


{\-x){\-2x) 


Hence  ^=  -  (11  -  14a;)(l -a;)2(l -2^)=^. 

„     -p.      _(aa:  +  6)^(ca;  +  c?)g     1  dy^_    ap  qc 


(ex+fy  y  dx    ax  +  h    cx  +  d    ex+f 

4a:3 


5     If  V=:     h  +  2bx  +  cx^  dy _  h{a-cx'^ 

yi  a  -  2hx  +  cx^^         dx     /..     m..  .    _'>x4/„  , 


dx    \/{l+x*){\-x*r 
h{a-c 
{a  -  2bx  +  cx'^)^  {a  +  2bx  +  cx^)'^ 


§37.    Important  Example. 

If  y  =  e~'"'  sin  bx, 


-^  =  sin  bx  -^  {e~'"')  +  e""""  -r-  (sin  bx) 
=  e""*  {-asmbx  +  b  cos  bx). 


PAETIAL  DIFFEEENTIAL   COEFFICIENTS  71 

Now  if        a  =  tan~i(-V  a  and  h  being  positive, 
cos  a  =  -7 and 


Therefore  -~-=  -  Ja^  +  //^  e~"''(sin  ire  cos  a  -  cos  Ja;  sin  a) 

ax  ^  ' 

=  _  ^^2-:^2  e-«-  sin  {bx  -  a). 
Thus  the  tangent  to  the  curve  y  =  e~"'''smbx  is  parallel  to  the 
axis  of  2;,  when  bx  =  n7r  +  a, 

and  the  equation  defines  an  oscillating  curve  with  continually 
diminishing  amplitude  in  the  waves  as  we  proceed  along  Ox. 
It  is  easy  to  show  that  when 

y  =  e""*  sin  {bx  +  c), 
-^  =  ^a-  +  h-  e"""  sin  (bx  +  c  +  a), 

UjX 

and  that  here  the  waves  increase  in  amplitude.     Corresponding 
results  hold  for  the  case  of  the  cosine. 

§  38.  Maxima  and  Minima  Values  of  a  Function  of  one 
Variable. 

The  student  is  already  familiar  with  the  graphical  and 
algebraical  discussion  of  the  maxima  and  minima  of  certain 
simple  algebraical  expressions.  The  methods  of  the  Differential 
Calculus  are  well  adapted  to  the  solution  of  such  problems. 

If  the  graph  of  the  function  is  supposed  drawn,  the 
turning-points,  or  places  where  the  ordinate  changes  from 
increasing  to  decreasing,  or  vice  versa,  can  only  occur  where 
the  tangent  is  parallel  to  the  axis  of  x,  as  in  the  points 
Aj,  A2  .  .  .  of  Fig.  17,  or  where  it  is  parallel  to  the  axis  of  y  as 
in  the  points  Bj,  B^  .  .  .,  except  in  such  cases  as  the  points 
Cj,  C2  .  .  .,  where,  although  the  curve  is  continuous,  the  gradient 
suddenly  changes  sign,  without  passing  through  the  value  zero 
or  becoming  infinitely  great. 

In  case  (A) :  -j-  is  zero  at  the  turning-point ;  and  if  this  point 
ax 

is  one  at  which  the  curve  ceases  to  ascend  and  begins  to  descend, 
-^  changes  from  being  positive  just  before  that  point  to  being 


72    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS 

negative  just  after.  At  such  a  point  the  function  is  said  to 
have  a  irmximum  value.      In   the   other  case,   where  the  curve 

ceases  to  descend  and  begins  to  ascend,  -^  changes  from  nega- 
tive to  positive,  and  we  have  a  minimum.  In  Fig.  17,  at  A^ 
there  is  a  maximum ;  at  A^  there  is  a  minimum. 

In  case  B  :    ;7-  is  infinitely  great  at  the  turning-point ;   and 
at  B^,  where  there  is  a  maximum,  it  changes  from  positive  to 


Fig.  17. 

negative,  while  at  B^,  where  there  is  a  minimum,  it  changes  from 

negative  to  positive. 

The  other  turning-points,  C^,  0^  ^^^  ^ig-  ^^  correspond  to  dis- 

dii 
continuities  in  -p,  but  it  can  be  shown  that  these  will  not  occur 

in  the  functions  with  which  we  are  dealing. 

We  thus  obtain  the  following  rule  for  finding  the  maxima  and 
minima  of  a  function  f{x),  omitting  cases  B  and  C. 

Obtain  f(x)  and  solve  the  equation  f{x)  =  0.  Let  its  roots  he  x^ , 
^2,  ....  Examine  the  behaviour  of  f\x)  in  the  neighbourhood  of  each 
of  these  roots. 


PAETIAL   DIFFERENTIAL   COEFFICIENTS  73 

If  f'{x)  changes  from  positive  to  negative  as  ive  pass  through  one  of 
these  roots,  then  f{x)  has  a  maximum  value  there. 

Iff{x)  changes  from  negative  to  positive  as  we  pass  through  one  of 
these  roots,  then  f{x)  has  a  minimum  there. 

If  f'{x)  does  not  change  sign  as  we  pass  through  the  root  consideredy 
then  f(x)  has  neither  a  maximum  or  a  minimum  there.    . 

§39.    Points  of  Inflection. 

Although  the  vanishing  of     '    is  a  necessary  condition   for  a 

maximum  or  minimum,  it  is  not  a  sufficient  condition,  since  the 

gradient  of  the  curve  may  become   zero  without  changing  its 

sign  as  we  pass  through  that  point.     Examples  of  such  points 

are  to  be  found  in  D^,  D.^  of  Fig.   17.     In  the  case  of  D^,  the 

gradient  is  positive  before  and  after  the  zero  value  ;  in  the  case 

of  D2,  it  is  negative.    At  these  points  the  curve  crosses  its  tangent^ 

and  when  this  occurs,  whether  the  tangent  is  horizontal  or  not, 

the  point  is  called  a  point  of  inflection. 

From  what  we  have  already  seen  [cf.  §  18]  as  to  the  conclusions 

d-y 
we  can  draw  from  the  sign  of  v^,  it  is  clear  that,  as  we  pass 

through  the  point  D^  of  Fig.  17,  —^  changes  from  being  negative 

to  being  positive.     The  curve  is  convex  upwards  just  before  D^ : 

it  is  concave  upwards  just  after  Dj.     At  D^,  ;7 1  =  0. 

It  will  be  seen  on  drawing  a  figure  that  at  points  where 
a  curve  crosses  its  tangent,  the  second  differential  coefficient 
vanishes  and  changes  sign,  provided  that  the  gradient  of  the  curve 
is  continuous. 

It  is  also  easy  to  show  that  when  ^  =  0,  and  -t4  is  negative^ 
there  is  a  maximum. 

And  when  -/  =  0,  and  —  ^^  positive,  there  is  a  minimum. 

Ex.  1.  Show  that  y  =  ax'^  +  2bx  +  c  has  alwa3^s  one  turning-point;  and 
point  out  when  it  is  a  maximum  and  when  it  is  a  minimum. 

2.    Find  the  maximum  and  minimum  ordinates  of  the  curve 
y  =  jc^ -6x^+12, 
and  also  find  the  points  of  maximum  gradient. 


74    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS 

3.  Find  the  turning-points  of  the  curve  y^^ix+lfix-^)^,  and  show 
that  (  -  1,  0)  is  a  point  of  inflection, 

4,  Find  the  turniner-points  of    y  =  , ^,    and  of    y  =  — -,. 

^  ^  -^     {x-2)  ^     cx  +  d 

HO.  Partial  Differentiation. 

So  far  we  have  been  considering  functions  of  only  one  inde- 
pendent variable,  i/=f(x).  Cases  occur  in  Geometry  and  in  all 
the  applications  of  the  Calculus  where  the  quantities  which  vary 
depend  upon  more  than  one  variable.  For  instance,  in  Geometry 
the  co-ordinates  of  any  point  (x,  y,  z)  upon  the  sphere  of  radius  a, 
whose  centre  is  at  the  origin,  satisfy  the  relation 

Hence  we  have  z'^  =  a-  - .?:-  -  i/, 

and  if  we  cut  the  sphere  by  a  plane  parallel  to  the  yz  plane, 
a,long  the  circle  Avhere  this  plane  cuts  the  sphere  x  is  constant, 
a,nd  the  change  in  z  is  due  to  a  change  in  y  only.  In  the  section 
by  a  plane  parallel  to  the  zx  plane,  the  change  in  z  would  be  due 
to  a  change  in  x  only.     Similar  results  hold  for  other  surfaces. 

Again,  the  area  of  a  rectangle  whose  sides  are  x  in.  and  y  in. 
is  xy  sq.  in.,  and  we  may  imagine  the  sides  x  and  y  to  change 
in  length  independently  of  each  other;  while  the  volume  of  a 
rectangular  1)0x  whose  edges  are  x,  y,  and  z  in.  is  xyz  cub.  in., 
and  X,  y,  z  may  be  supposed  to  change  independently. 

The  ordinary  gas  equation 

pv 

7m  =  constant 

is  another  example  of  the  same  sort  of  relation,  and  it  would 
be  easy  to  multiply  these  instances  indefinitely. 

Let  the  equation  ^=f(x,  y) 

•express  such  a  relation  between  two  independent  variables  x  and 
y,  and  a  dependent  variable  z. 

Let  us  suppose  that  the  independent  variable  y  is  kept  constant 
and  that  x  changes. 

Then  the  rate  at  which  z  changes  with  regard  to  x,  when  y 
is  kept  constant,  will  be  given  by 

j^^  f/(.T4-ga;,  y)-f(x,  y)'\ 


PARTIAL   DIFFERENTIAL   COEFFICIENTS  75 

In  the  second  case  let  x  be  kept  constant  and  let  y  change. 
Then  the  rate  at  which  z  changes  will  be 


Lt  )f{^'^y+^y)-f{^,y) 


\ 


These  two  differential  coefficients  are  called  the  Partial  Differ- 
ential Coefficients  of  z  with  regard  to  x  and  y  respectively,  and 

are  written  .c-  and  ^"  respectively.* 

Ex.  1.    When  z  =  xy,  prove,  from  the  definition,  that  ^=y,  and  ^=-x. 

2.  When  2az  =  a:^  +  y^,  prove,  from  the  definition,  that  :=^  =  -,  and  r^-  =  -• 

^    -^  Ox    a'  dy    a 

3.  If  u  —  xyz,  prove,  from  the  definition,  that  .-^  =  yz. 

§41.    Total  Differentiation. 

When  the  variables  x  and  y  in '  the  above  examples  both 
depend  upon  a  third  variable  t,  z  will  vary  in  value,  as  x  and  y 
change  with  /. 

In  Ex.  1  above,  ^"^^V^ 

z-\-^z  =  {x  +  &)  {y  +  hj). 
8z        8x        8y     8x 
bt      -^  hi  bt       bt    ■' 

Proceeding  to  the  limit,  we  have 

dz        dx        dy 
*       Tt^-'~dt^^ti' 

But  y'^'?r  ^^*^  ^  ^  ^'  ^^^"  ^  ^  ^y- 

Therefore,  in  this  case, 

dz  _  dz  dx     dz  dy 
dt     dx  dt     dy  Hi' 

In  Ex.  2  above,  laz^x'^  +  y-, 

we  find  'la^z  =  2x8x  +  2y8y  +  (8xy  +  (8yY. 

*Itishardly  necessary  to  point  out  that  this  symbol  —  stands  for  an  operation, 

Ox 
and  that  dz,  dx  are  not  to  be  considered  separately ;  also  that  this  is  a  different 
notation  from  the  5x  of  our  earlier  work. 


76    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS 

rpi  dz        dx        dy 

,,    ,         .  dz     dz  dx      dz  dy 

sothataga.n  _  =  _  ^  +  _ -^. 

It  can  be  shown  that  this  holds  in  general,  but  the  proof  of  the 
theorem  cannot  be  taken  at  this  stage  of  our  work. 

The  differential  coefficient  -f^  is  called  the  Total  Differential 

at 

Coefficient  in  such  cases,  as  compared  with  the  Partial  Differen- 
tial Coefficient  defined  above. 

As  a  special  case,  when  z  =f{x,  y)  and  y  is  a  function  of  x,  we 
obtain  ^  _  ?^     '^di 

dx     dx     dy  dx^ 
and  the  left-hand  side  is  called  the  Total  Differential  Coefficient 
of  z  with  regard  to  x. 

Also  the  result  that,  when  z=f(x,  y)  and  .r,  y  are  functions  of  ^, 

dz  _  dz  dx      dz  dy 
dt~ dx  dt      dy  dt 

may  be  used  to  obtain  an  approximation  to  the  small  change  hz 

in  z  due   to  the  small  changes  ^x  and  %  in  x  and  y,  when  t 

becomes  t  +  U. 

For,  as  we  have  already  seen  (p.  23), 

dx 
8x  will  be  approximately  -jr  8t ; 

dy 
Sy  will  be  approximately  -jj  8t ; 

dz 
and  8z  will  be  approximately  -ji  8f. 

We  thus  have,  on  multiplying  the  above  equation  by  8t, 
8z  =  7s-8x  +  ~  8y,  approximately. 

§42.    Differentials.* 

In  the  case  of  the  curve  y=f{x),  the  increment  8y  of  y  which 
corresponds  to  the  increment  8x  of  x,  is  given  in  Fig.  18  l)y  HQ. 

*  §  42  may  be  omitted  on  first  reading. 


PARTIAL  DIFFERENTIAL   COEFFICIENTS 


77 


Also 


HQ  =  HT  +  TQ=:Sa:^  +  TQ 

.-.  5^  =  ga.g  +  TQ. 


As  Sx  gets  smaller  and  smaller,  TQ  gets  smaller  and  smaller, 
at  least  in  the  neighbourhood  of  P. 

The    "  small    quantity "    TQ  is   a    smaller    "  small   quantity " 
than  &r,  since  8y  _dy     TQ, 

8x    dx      Sic ' 

and  in  the  limit  -^  is  equal  to  -f~,  so  that  — -^  must  disappear  in 
^,     ,.    .^  &  dx  8x 

the  limit. 

In  mathematical  language,  if  8x  is  an  infinitesimal  (or  small 
quantity)  of  the  first  order,  TQ 
will  be  at  least  an  infinitesimal 
of  the  second  order. 

It  is  convenient   to    have  a 
name     and     symbol     for     this 

quantity     -^  8x. 


The 


adopted  is  the  "differential  of 
y,"  and  the  symbol  \s'"dy." 

Hence  with  this  definition  of 
the  term  "  differential," 


^^-m 


8x, 


M 

Fig.  18. 


where  we  have  enclosed   -^  in  brackets  on  the  right-hand  side, 

so  that  it   may  be  clear    that  this   stands  for  the   differential 
coefficient  obtained  by  the  processes  we  have  been  developing 
in  the  preceding  pages. 
By  the  above  definition 

d{f{x))=^f'(x)8x,    where    f'{x)  =  f^; 
and  dx  =  8x. 

So  that  dy  =f\x)  dx^     when     y  =f{x). 

Hence  we  may  restate  our  definition  as  follows  : — 

The  differential  of  the  independent  variable  is  the  actual  imyrement 
of  that  variable. 


78    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS 

The  differe7itial  of  a  function  is  the  differential  coefficient  of  the 
function  midtiplied  by  the  differential  of  the  iridependent  variable. 

In  this  definition  it  is  not  necessary  to  assume  that  the 
differentials  are  small  quantities  or  infinitesimals,  but  in  all 
the  applications  of  this  notation  this  assumption  is  made. 

Then  the  equation  dy  ^f{x)  dx 

will  give  the  increment  of  y,  if  small  quantities  of  the  second 
order  be  neglected. 

Such  an  equation  as  dy=f'(x)dx, 
a  differential  equation  as  it  is  called,  may  be  used  to  give  the 
approximate  change  in  the  dependent  variable,  and  from  this 
point  of  view  it  saves  the  trouble  of  writing  down  the  equation 
between  the  increments,  and  then  cutting  out  the  terms  which 
are  so  small  that  they  may  be  neglected. 

Ex.  1.  Write  down  a  table  of  differentials  corresponding  to  the 
standard  diflferential  coefficients. 

e.g.  d{x^^)  =  nx"-'^dx. 

2.  If  a:  =  a  cos  6,  y  =  asm  6,  prove  b}'  differentials  that  -j-—-  cot  d. 

3.  If  a;  =  a(w<  +  sin  w<),  y  —  a{\-Q,o%wt)^  prove  that -^  =  -         ^ 


dx     1  +  cos  o)t 
4.    If  2  =  xy,  prove  that  dz  =  r^dx  +  T^  dy. 

EXAMPLES   ON  CHAPTER  V 

1.  Find  the  differential  coefficients  of 

(i.)  a;e^,     (ii.)  ar^'e''^     (iii.)  (ax  +  h)e''''+^,     (iv.)  e**i»~\ 

2.  Find  the  differential  coefficients  of 

(i.)  ei+^^     (ii.)  xV-^,     (iii.)  a;'»e«*",     (iv.)  a;"»a^". 

3.  Find  the  differential  coefficients  of 

(1.)  x-logx,    (ii.)log('l^^V     (iii.)log(x/^l+V^^,     (iv.)log('L^'V 
(v.)log(        JL      \    {Vi.)\og(\±-^f). 

4.  Differentiate  the  following  expressions  logarithmicallj^ : — 

(i.)  x/(2x-+l)(.r-2),     (ii.)     ^  (iii.)  ^         ,     (iv.)  a-, 

,     >   sin"?na;      /   •  \   /,      1\* 
(v.)  — - — ,     (vi.)      1  +  - )  ; 
cos"*  ?i.r  \       xj 

and   point   out   why  we   cannot   apply   our   formula   for   the   differential 

coefficient  of  x^  to  the  case  of  yf. 


{x<a<§) 

dy 

sih-  -  ii' 

dx 

a  +  b  cos  X' 

(62  >  a2) 

dr 

1 

PARTIAL  DIFFERENTIAL   COEFFICIENTS  79 

5.  If  y=   ,    ^       tan-if  ^"^V  prove  that  ^  =  -^^4j («c>62) 

6.  Ify  =  hogg+ii%-Ltan-(-^),  prove  that  ^^^^   /^-,. 

7.  If  y  =  2cos-^\^~-|,  provethat  ^=    ,          ^  (a>;c>^) 

o     Ti-  2  ,      la  -  X  ^y    ^  dy  1 

8.  If  V  =  -T"-     cos-^  \/ ,  prove  that  -r-  = .  • 

9.  it  y  —  \os[ i I,  prove  that 

10.  If  r  —  -■ tan-^  {  \  j-tan^V,  prove  that  -^  = 

\/a2-62  \  \a  +  6         2/'  ^  dd     a  +  b  cos  6 

(a2>62) 

11.  In  the  curves  whose  equations  in  polar  co-ordinates  are 

(i.)  r  =  ae^cotaj       (ii.)  ?-"  =  a"sin?i^,       (iii.)  r"  =  a"cosw^, 
(iv.)  r"  =  a"secn^,       (v.)  7^"  =  a" cosee 7i^, 

df) 
find  r  — .     Can  you  give  any  geometrical  meaning  to  this  expression  ? 

12.  If  2/  =  e-^sin(2x+l),  prove  that  ^  =  2V2.e-2^cos  f  2.C+ 1  +  jY 

13.  Find  the  value  of  -^  in  the  following  curves ;   discuss  the  way  in 

which  it  changes  as  x  passes  along  the  axis  ;  and  find  the  turning-points, 
if  there  are  any,  of  each  curve  : — 

{i.)  y  =  x{x-ir. 
(ii.)  y  =  x'^{x-iy\ 
(iii.)  y  =  {x-l)Hx-2f. 
,.     ,         x'^  +  x+l  • 

,     ,  a;2  -  a;  +  1 

(v.)  y 


(VI  )  y  = 

(vii.)  y  = 

(viii.)  y  = 

(ix.)  y  = 

(x.)  y  = 


x^  +  x+1 
(a;-l)(a;-2) 
~x'^  +  x+l    ' 

X'^  +  X+l 

{x-\)[x-2) 
{x-\){x-^) 
(a;-2)(aj-4) 
{x-\)(x-2) 

ar^+1 


x^ 

[These  curves  are  discussed  algebraically  and  drawn  to  scale  in  Chrystal's 
Introduction  to  A/gebra,  pp.  391-404.  The  student  is  recommended  to 
compare  his  results  with  those  to  be  deduced  from  these  figures.] 


pa8d_\ 


SO    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS 

14.    If  2  =    o  +  r:7,  prove  that  x^^  -^y  :r^—^z. 
a?     h-    ^  ^x     ^  dy 

16.  The  formula  for  the  index  of  refraction  ;u  of  a  gas  at  temperature  Q" 
'  and  pressure  ^9  is  fx^-\      p 

where  /io  =  the  index  of  refraction  at  0°, 

a  =  the  coefficient  of  expansion  of  the  gas. 
Prove  that  the  effect  of  small  variations  M  and  5^  of  the  temperature 
and  pressure  on  the  index  of  refraction  is  to  cause  it  to  vary  by  an  amount 

^      760  Vl  +  a^    (1 

17.  If  'pv  —  R^  is  the  ordinary  gas  equation,  where  Q—\-\-at,  write  down 
the  values  of 

(U.)  |. 

(iii.)  The  approximate  increase  in  the  pressure  due  to  a  small  decrease 
in  the  volume,  the  temperature  being  unchanged, 

(iv.)  The  approximate  increase  in  the  volume  due  to  a  small  increase  in 
the  temperature,  the  pressure  remaining  the  same, 

(v.)  The  approximate  increase  in  the  pressure  due  to  a  small  increase 
in  both  temperature  and  volume. 

18.  Assuming  that  the  H.P.  required  to  propel  a  steamer  of  a  given 
design  varies  as  the  square  of  the  length  and  the  cube  of  the  speed,  prove 
that  a  :4%  increase  in  length,  with  a  7  %  increase  in  H.P.,  will  result  in 
a  I  %  increase  in  speed. 

19.  The  area  of  a  triangle  is  calculated  from  measurements  of  two  sides 
and  their  included  angle.  Determine  the  error  in  the  area  arising  from 
small  errors  in  these  measurements.  ^ 

20.  Assuming  that  the  area  of  an  ellipse  whose  semiaxes  are  a  and  h 
inches  is  -Kah  sq.  in.,  and  that  an  elliptical  metal  plate  is  expanding  by  heat 
or  pressure,  so  that  when  the  semiaxes  are  4  and  6  inches,  each  is  increasing 
at  the  rate  *i  in.  per  second,  prove  that  the  area  of  the  plate  is  increasing 
at  the  rate  of  tt  sq.  in.  per  second. 


CHAPTER  VI 

THE  CONIC   SECTIONS*       ^ 

§  43.    Introductory. 

In  this  chapter  we  shall  very  briefly  examine  the  properties 
of  the  Conic  Sections,  or  the  curves  in  which  a  plane  cuts  a 
Right  Circular  Cone.  It  is  shown  in  the  Geometry  of  Conies 
that  these  curves  are  the  loci  of  a  point  which  moves  in  a  plane 
so  that  its  distance  from  a  fixed  point  is  in  a  constant  ratio  to 
its  distance  from  a  fixed  straight  line.  The  fixed  point  S  is 
called  the  focus ;  the  fixed  line,  the  directrix ;  and  the  constant 
ratio,  e,  the  eccentricity. 

When  e<l,  the  curve  is  called  an  Ellipse ; 

when  g=  1,  the  curve  is  called  a  Parabola ; 
when  e>l,  the  curve  is  called  a  Hyperbola; 
and  the  circle  is  a  special  case  of  the  ellipse,  the  eccentricity 
being  zero,  and  the  directrix  at  infinity. 

§44.    The  Parabola  (e=l). 
(i.)  To  find  its  equation. 

Let  the  focus  S  be  the  point  (a,  0),  and  the  directrix  the  line 
x  +  a  =  0  (Fig.  19). 

Let  P  be  the  point  (x,  y). 
Then  since  SP2  =  PM2, 

(x-af  +  y^  =  (x  +  af. 
,'.  y^  =  4cax. 

*  The  student  is  referred  for  a  fuller  discussion  of  the  properties  of  the  Conic 
Sections  to  the  books  mentioned  on  p.  15.  Many  of  their  properties  are  most 
easily  obtained  geometrically,  and  are  to  be  found  in  books  on  Geometrical 
Conies. 

C.C.  F 


82 


THE  CONIC  SECTIONS 


This  is  the  equation  of  the  parabola  with  the  origin  at  the 
point  where  the  curve  cuts  the  perpendicular  from  S  on  the 
directrix.     This  point  is  called  the  vertex  of  the  curve ;  the  axis 


M 

1 

P^^ 

1 

1          L 

Y 

.      X 

i 

S               N 

1 
1 
1 
1 

1 

L 

^ 

Fig.  19. 


of  X  is  called  the  axis  of  the  curve ;   and  the  ordinate  L'SL 
through  the  focus  is  called  the  Latus  Rectum. 

(ii.)  The  shape  of  the  curve. 

From  the  form  of  the  equation  of  the  curve  we  see  that  the 
curve  lies  wholly  to  the  right  of  the  axis  of  y,  and  that  it  is 
symmetrical  with  regard  to  the  axis  of  x.' 


Also  since 


2'a-=2«' 


dy  _^a  _    la 
dx     iJ  ~  y  X 


~,  when  ^>  0. 


It  follows  that  the  tangent  at  the  vertex  coincides  with  the. 
axis  of  y,  and  that  as  we  move  along  this  branch  of  the  curve 
in  the  direction  of  x  increasing,  the  curve  continually  ascends, 
the  slope  getting  less  and  less  the  greater  x  becomes. 


THE  CONIC  SECTIONS  83 

(iii.)  The  equations  of  the  tangent  and  normal  at  {x^,  y^). 

du  2fl 

Since  the  value  of  ~  at  (Xq,  y^  is  — ,  the  equation  of  the 
ax  jQ 

tangent  there  is 

x-x^     2/0' 
or  y^{y-y^)=^2a{x-XQ). 

This  becomes  yyQ  —  2a{x  +  x^,  since  y^  =  iaxQ. 
Also  the  normal  is  the  line 

yQ{x-x^)-\-2a{y-y^)  =  0, 
since  this  line  passes  through  {xq,  y^  and  is  perpendicular  to 
the  tangent. 

EXAMPLES  ON  THE   PARABOLA 
L    Show  that  the  curves  a;^=  ±4i/  are  parabolas,  and  plot  the  curves. 

2.  Show  that  the  equation  y  =  ax^  +  2hx  +  c  always  represents  a  parabola, 

and  plot  the  curves 

(i.)     y  =  x^  +  ^X  +  ^, 

(ii.)  Ay  =  x'^  +  4LX-^, 
(iii.)     x  =  y'^^y. 
Find  also  (i.)  The  co-ordinates  of  their  foci ; 

(ii.)  The  co-ordinates  of  their  vertices  ; 
(iii.)  The  equations  of  their  latera  recta  ; 
(iv.)  The  lengths  of  their  latera  recta  ; 
(v.)  The  equations  of  their  axes  ; 
(vi.)  The  equations  of  the  tangents  at  their  vertices. 

3.  Find  algebraically  and  graphically  the  minimum  value  of  the  expres- 
sion a;2-2a;-4,  and  the  maximum  value  of  b  +  Ax-2x^. 

4.  The  tangent  at  P  meets  the  axis  of  the  parabola  of  Fig.  19  in  T,  and 
the  normal  meets  the  axis  in  G.     Prove  the  following  properties  : — 

(i.)  AN  =  AT, 

(ii.)  SP  =ST  =  SG, 

(iii.)  NG  =  2AS, 
and  show  that  the  tangents  at  the  ends  of  a  focal  chord  meet  at  right 
angles  on  the  directrix. 

5.  Prove  that  the  line  y  =  x-\-\  touches  the  parabola  y'^  =  4rX,  and  that 
the  line  y  =  mx-\ —  touches  the  parabola  y^  =  4iax.  Find  the  point  of 
contact  in  each  case. 

6.  Find  the  equations  of  the  tangent  and  normal  at  the  point  where 
the  line  x  =  2  cuts  the  parabola  x^='iy. 


84 


THE  CONIC  SECTIONS 


7.  Find  the  equations  of  the  tangents  and  normals  at  the  extremities 
of  the  latus  rectum  of  the  parabola  'ip'  =  ^x,  and  show  that  they  form  a 
square. 

8.  Prove  that  the  locus  of  the  middle  points  of  the  chords  of  the 
parabola  y^=^ax,  which  make  an  angle  Q  with  the  axis  of  x,  is  the 
straight  line  y^^acotd. 

9.  The  chord  PQ  meets  the  axis  of  the  parabola  of  Fig.  19  in  0. 
PM  and  QN  are  the  ordinates  of  P  and  Q.  Prove  that  AM.  AN  =  A02, 
by  finding  the  equation  of  the  chord  in  its  simplest  form. 

10.  The  position  of  a  moving  point  is  given  by  the  equations 

x  =  vcos,a.t, 
y  =  vs,v[ia.t-\gt^. 
Interpret  the  equations,  and  prove  that  the  point  moves  on  a  parabola 
whose  axis  is  parallel  to  the  axis  of  y  ; 

/  if^  sin  a  cos  a     'i^  sin'^a \ 


whose  vertex  is  at  the  point 
whose  directrix  is  the  line  y  ■ 


2.9 


and  whose  latus  rectum  is  of  length 

§45.   The  Ellipse  (e<l). 
(i.)  To  find  its  equation. 


2v^  cos^  a 
9 


M 


/ 

P 

r 

B 

-, 

\ 

-I 

/ 

s 

c 

B' 

S' 

t- 

X' 


Fio.  20. 


Let  the  axis  of  x  be  the  axis  of  the  ellipse  (i.e.  the  line  through 
the  focus  perpendicular  to  the  directrix). 
Let  S  be  the  point  (c?,  0) ; 

and  let  the  axis  of  y  be  the  directrix. 


THE   CONIC  SECTIONS  85 

Let  P{x^  y)  be  any  point  upon  the  curve. 
Then  SP2  =  e2pM2. 

...  x'^iA  -e^)-2xd  +  y'^=  -d'^. 

/ ^\2        ^  d'^  d^  d^e^ 

•'•   V     l-eV  +r-e2-(i_g2y2     i  _72-(i  _g2)2- 

Now  change  the  origin  to  the  point  (r— 2'  ^)»  keeping  the 
axes  parallel  to  their  original  directions. 
The  equation  of  the  ellipse  then  becomes 

^2  ^2^2 


a;2  «/2 

i-e.  rrrr— 4-       i^     =  1. 


(l-e2)       (l-e2)2' 

,2  y2 


Putting  a2  = 

and  ^>2  = 


(1  -  e2)2     (1  _  e2) 
ri2e2 


(1  -  ^2)2 

d^ 
1  -  «2' 


/Tr'2  .1*2 

we  have  l  +  f^'^-^'     where     ^2  =  0^2^1  _g2^. 

In  this  form  the  origin  C  is  called  the  centre  of  the  curve, 
since  it  bisects  every  chord  which  passes  through  it.     This  is 

a;2     y2 
clear,  since  if  {x^,  y^)  lies  on  -2  +  T2=  1>  so  does  {-x^,  -Vi)- 

d  de^ 

Also  we  notice  that  CS  = .; 5  -d=  , — —.  =  ae, 

and  that  CX=--^  =  -. 

I  -  e2     g 

From  the  symmetry  of  the  equation 
a;2     y2 
a2  +  62-J' 

it  is  clear  that  there  is  another  focus,  namely,  the  point  {ae,  0) ; 

and  another  directrix,  the  line  aj  =  -,  with  regard  to  the  axes 
through  the  point  C. 


86  THE  CONIC  SECTIONS 

The  axis  of  x  is  in  this  case  called  the  major  axis,  and  the 
axis  of  y  the  minor  axis.  The  one  is  of  length  2a ;  the  other  of 
length  2h.  If  h  had  been  greater  than  a,  the  foci  would  have 
lain  upon  the  axis  of  y,  and  this  axis  would  have  been  the  major 
axis.     When  a  and  h  are  known,  the  eccentricity  e  is  given  by 

In  the  circle  a—h^  and  e  =  0. 

(ii.)  The  shape  of  the  curve. 

Since  the  equation  involves  only  the  terms  x'^  and  y^,  the 
curve  is  symmetrical  about  both  the  axes  of  x  and  y. 

1  — 2 )'  ^®  s®®  ^^^^  ^  must  lie  between 

-  a  and  +  a,  and  that,  as  x  passes  from  -  a  to  -\-a,  the  positive 
value  of  y  gradually  increases  from  zero  to  b,  and  then  diminishes 
again  to  zero. 

The  curve  is  thus  a  closed  curve,  lying  altogether  within  the 
rectangle  x=  ±a,  y=  ±h. 

This  is  also  evident  from  the  property  of  Ex.  3,  p.  87,  where  it 
is  proved  that  the  curve  may  be  drawn  by  fixing  the  two  ends  of 
a  string  of  length  la  to  the  points  S  and  S',  and  holding  the 
string  tight  by  the  point  P  of  the  tracing  pencil. 

(iii.)  The  equations  of  the  tangent  and  normal  at  (Xq,  y^). 

Since  +1^=1 

a^'^¥  dx~ 
Therefore  the  equation  of  the  tangent  at  {x^,  y^)  is 

x-Xq         a%' 

X  v 

which  becomes        (x  -Xo)-^  +  {y-  yo)  )^  =  ^  '> 

^^(i    VVn     1       •         ^(?    y^     -I 


THE  CONIC  SECTIONS  87 

It  follows  that  the  equation  of  the  normal  is 

(»:-^o)|-(3'-%)|  =  0. 


^0      Vo 


EXAMPLES  ON  THE  ELLIPSE 

1.  Trace  the  ellipses    (i. )  Zx^  +  4y2=  12  ; 

(ii.)  3(x-l)2  +  4(y-2)2=zl2; 
(iii.)  a:2  +  4y2  =  8y; 
(iv.)  4a;2  +  3y2=zl2; 

und  find  the  co-ordinates  of  the  foci  and  of  the  extremities  of  the  axes,  the 
length  of  the  latus  rectum,  and  the  eccentricity  of  each. 

2.  In  the  ellipse  -5  +  ^  =  1,  show  that  the  co-ordinates  of  any  point 


62 


may  be  expressed  as  a;  =  a  cos  5,  y=&sind;  and  interpret  the  result 
geometrically. 

3.  P    is    the    point    (a^i,  y{)    on   the   ellipse   ^  +  p  =  l.      Prove   that 

SP  =  a  +  ea:i  and  S'P^a-eXj,  and  deduce  that  the  curve  is  the  locus  of 
A  point  which  moves  so  that  the  sum  of  its  distances  from  two  fixed 
points  is  constant. 

4.  The  tangent  at  P  meets  the  major  axis  in  T,  and  PN  is  the  ordinate 
of  P  ;  prove  that  CN  .  CT  =  C A2. 

5.  The  normal  at  P  meets  the  major  axis  in  G.  Prove  that  SG :  SP  =  e, 
And  deduce  that  PG  bisects  the  angle  SPS'. 

6.  Prove  that  the  middle  point  of  the  chord  y  =  x+\  lies  upon  y=  — -^x, 

and  that  the  middle  points  of  chords  parallel  to  y  =  mx  lie  upon  the  chord 

62 
'u  =  7n'x,  where  mm'  +  — ,  =  0. 

7.  If  CP  bisects  chords  parallel  to  CD,  prove  that  CD  bisects  chords 
parallel  to  CP  (CP  and  CD  are  then  said  to  be  conjugate  diameters) ;  and 
j)rove  that  the  tangents  at  P  and  D  form  with  CP  and  CD  a  parallelogram. 

8.  If  P  is  the  point  (a  cos  6,  6  sin  6),  prove  that  CD  is  the  line 

a  sin  dy  +  h  cos  dx  =  0, 
4ind  deduce  that  CP2  +  CD2  =  a2  +  h'\ 


88  THE  CONIC  SECTIONS 

§46.    The  Hyperbola  (e>l). 

(i.)   Tojiiid  its  equation. 

Proceeding  as  in  §  45  (i.),  we  obtain  the  equation 


where  we  have  written  a^  for 


a'     62 


and  62  for  ^^\,    i.e.  for  a2(g2  _  i). 

Also  d  is  the  distance  from  the  focus  S  to  the  directrix. 


It  follows  that  CS  =  fte,  CX  =  -,  and  that  there  are  two  foci 
and  two  directrices. 

The  line  joining  the  foci  S,  S'  is  called  the  transverse  axis  of 
the  hyperbola. 

(ii.)  The  shape  of  the  curve. 

The  form  of  the  equation  shows  that  the  curve  is  symmetrical 

about  both  axes.     Also  since  i/^h^l—^-lX  it  is  clear  that  x 

cannot  lie  between  -  a  and  +a  ;  since  x^  =  a^ll  +'j»),  y  can  have 
any  value  whatsoever. 


THE  CONIC  SECTIONS  89 

If  we  write  the  equation  as 

we  see  that,  when  x  is  numerically  very  great,  ^  is  less  than,  but 

very  nearly  equal  to   -2;  and  that  for  all  points  on  the  curve 

^  is  less  than  — . 
x^  a^ 

Also  the  positive  value  of  y  decreases  as  x  losses  from  -  qo  to 
-  a,  where  it  vanishes ;  and  it  increases  w^ithou"Mimit  from  the 
value  zero  at  a;  =  a,  as  a;  passes  along  the  positive  axis  of  x. 

The  shape  of  the  curve  is  thus  as  in  Fig.   21.      The  lines 

y=  ±~  X  are  called  the  asymptotes,  and  the  curve  lies  wholly 

between  those  lines ;  while,  as  the  numerical  value  of  x  gets 
greater  and  greater,  it  approaches  more  and  more  nearly  to  these 
lines,  without  ever  actually  reaching  them. 

(iii.)  The  equations  of  the  tangent  and  normal  at  (Xq,  y^)  are  easily 
shown  to  be 


XX, 


a2      62 

and  |(^_^^)  +  |Q(y_y^)  =  0. 

(iv.)  The  'product  of  the  perpendiculars  from  any  j^oint  on  the 
curve  to  the  asymptotes  is  constant. 

The  asymptotes  are  the   lines  y  =  ±-x.      Then  if   PM,  PN 

are  the  perpendiculars  to  these  lines  from  the  point  (a^^j,  y^), 

h 

PM=— ;=^^,         PN  = 


V'"-l  - 


Therefore  PM  .  PN  =      "    .  ,/^  =  ^-^ , 

since  ^_^  =  1. 

a^      b^ 

Hence  PM  .  PN  =  constant. 


90  THE  CONIC  SECTIONS 

When  b'^  =  a^,  the  asymptotes  are  at  right  angles,  and  the 
eccentricity  is  ^^2.  In  this  case,  by  taking  the  asymptotes  as 
axes,  the  equation  x^-y^  =  a^  is  transformed  to 

2xy  =  a?. 

This  equation  is  of  the  form  xy  =  c^,  a  relation  which  is  of  the 
greatest  importance  in  Physics.  We  could  obtain  an  equation 
of  the  same  form  for  any  hyperbola  referred  to  its  asymptotes 
as  oblique  axes. 


EXAMPLES  ON  THE  HYPERBOLA 

1 .  Trace  the  hyperbolas  : 

(i.)  3a:2-4v2=12, 
(ii.)  3(a:-l)2-4(y-2)2=12, 
(iii.)  x^-Ay'^  =  %y, 
(iv.)  4a;2_3y2^i2; 

and  find  the  co-ordinates  of  the  foci  and  of  the  points  where  each  curve 
cuts  its  transverse  axis,  the  length  of  the  latus  rectum,  and  the  eccentricity 
■of  each. 

2.  Trace  the  rectangular  hyperbolas  : 

(i.)  xy=±4, 

(ii.)  2/=l±^., 

And  find  the  co-ordinates  of  the  foci  and  of  the  points  where  the  transverse 
axis  meets  each  curve. 

3.  Prove  that  the  tangent  at  [x^,  yo)  to  ^^e  hyperbola  xy  =  c^  is 
^o-hyXo  =  2c^,  and  that  the  point  of  contact  bisects  the  part  of  the  tangent 
cut  off  by  the  asymptotes. 

4.  P  is  the  point  (ccj,  y-^)  on  the  hyperbola  whose  equation  is  -:^-j^  =  \. 

Prove  that  SP  =  exi-a,  and  ^'V  =  ex-^  +  a,  and  deduce  that  the  curve  is  the 
locus  of  a  point  which  moves  so  that  the  difference  of  its  distances  from 
two  fixed  points  is  constant. 

5.  The  tangent  at  P  on  the  hyperbola  -2-^  =  1  meets  the  transverse 
axis  in  T,  and  PN  is  the  ordinate  of  P.     Prove  that  CN .  QT  =  a". 

6.  The  normal  at  P  meets  the  major  axis  in  G  ;  show  that  SG  =  eSP, 
and  deduce  that  PG  bisects  the  angle  SPS'. 


THE  CONIC  SECTIONS  91 

7.  Prove  that,  in  the  hyperbola  ^-|i  =  l,  the  middle  point  of  the 

1)2 

chord  y  =  x+l  lies  upon  the  line  y  =  —^x,  and  that  the  locus  of  the  middle 
points  of  chords  parallel  to  y  =  mx  is  the  line  y  =  mx,  where  mm  =-2- 

8.  If  CP  and  CD  are  two  conjugate  diameters  of  the  hyperbola  -j  -  f2  =  1 


a" 


(i.e.  if  each  bisects  chords  parallel  to  the  other),  prove  that  if  P  lies  upon 
this  curve,  CD  does  not  meet  the  curve,  and  that  if  D  is  the  point  where 

CD  meets  the  hyperbola  —^-p^^  -  1 , 

Cp2_CD2=a2_2,2, 


CHAPTER  VII 

THE   INTEGRAL   CALCULUS— INTEGRATION 

§  47.   Introductory. 

In  considering  the  motion  of  a  point  along  a  straight  line,  we 
saw  that,  if  ^  ^  y/ a 

is  the  relation  between  the  distance  and  the  time,  the  velocity  v 
is  given  by  ^, 

^  In  general,  the  problem  of  the  Differential  Calculus  is  as 
follows :  given  the  law  in  obedience  to  which  two  related 
magnitudes  vary,  to  find  the  rate  at  which  the  one  changes 
with  regard  to  the  other.  The  problem  of  the  Integral  Calculus 
is  the  inverse  one  :  given  the  rate  at  which  the  magnitudes 
change  with  regard  to  each  other,  to  find  the  law  connecting 
them.  In  other  words,  in  the  Differential  Calculus  we  determine 
the  infinitesimal  change  in  the  one  magnitude  which  corresponds 
to  an  infinitesimal  change  in  the  other,  when  we  know  what 
function  the  one  is  of  the  other.  In  the  Integral  Calculus  we 
determine  what  function  the  one  is  of  the  other,  when  the 
corresponding  infinitesimal  changes  are  known.  We  have  thus 
to  find  the  function  of  x,  denoted  by  y,  which  is  such  that 

The  value  of  y  which  satisfies  this  equation  is  written  |/(a;)c?a;, 

and  is  called  the  integral  of  f{x)  with  regard  to  x.  When  we  have 
found  the  integral  of  f(x),  we  are  said  to  have  integrated  the 
function.    The  process  of  finding  the  integral  is  called  integration. 


THE  INTEGRAL  CALCULUS— INTEGRATION      93 

/goa:                      ^    goa; 
e«*dr=— ,  since  ^ =e«^. 
a                eta;  a 

3.   Fill  up  the  blanks  in  the  following  table  : — 


/(^) 

1 

X' 

x^ 

ar* 

X* 

x^ 

a;io 

x" 

jf{x)dx 

n  being  a  positive  integer. 

4.  Fill  up  the  blanks  in  the  following  table  : — 


/(^) 

1 

a; 

1 
a:2 

1 
ar* 

1 

X* 

1 

1 

I 

jf{x)dx 

n+1  being  a  positive  integer. 
5.    Verify  the  following  results 


/(^) 

sin  a; 

cosa; 

tan  X     !     cot  X 

sec  a; 

cosec  X 

JAx)dx 

-cos  a; 

sin  a; 

log  sec  X    log  sin  a; 

i 

logtan^l  +  ^j 

log  tan  ^ 

6.    Verify  the  following  results  : — 


/(^) 

sin^a: 

cos^a; 

tan% 

cot^a; 

sec^a; 

cosec^a; 

jf{x)dx 

X    Bin  2x 
2        4 

X    sin  2a; 

2"^     4 

tan  a;  -  a; 

-  cot  x-x 

tana; 

-cot  a; 

In  each  of  these  cases  we  might  have  added  any  constant  to 
the  answer,  since  the  differential  coefficient  of  a  constant  is  zero, 
and  the  complete  result  in  the  first  two  examples  would  have 
^een  (    dx      ,      ,        ,     ^ 


a        ' 


where  C  is  called  the  constant  of  integration. 


94      THE  INTEGRAL   CALCULUS— INTEGRATION 

It  is  thus  evident  that  the  equations 

|f(x)=/(.) 


and 


Y(z)  =  ^f{x)dx 


represent  the  same  thing,  and  that  the  fuller  statement  of  the 
second  would  be 

F{x)  +  C=\f{x)dx. 

Owing  to  the  presence  of  the  arbitrary  constant,  l/(a;)6?a;  is 
called  the  Indefinite  Integral  of  f{x).  •' 

The  geometrical  meaning  of  the  constant  of  integration  is  that 

there  is  a  family  of  curves  all  having  the  same  slope  as  a  given 

curve.     The  curves  „  _ 

y^F{x)  +  C 


are  all  parallel,  when  C  is  given  different  constant  values. 


§  48.   Table  of  Standard  Integrals. 

When  integration  is  regarded  in  this  way,*  the  first  thing  we 
have  to  do  is  to  draw  up  a  list  of  the  most  important  integrals. 
This  table  is  obtained  from  the  corresponding  results  in 
differentiation.  Any  result  in  integration  can  always  be  verified 
by  differentiation.  Later  we  shall  see  that  there  are  certain 
general  theorems  on  integration  which  correspond  to  the  general 
theorems  of  differentiation.  These  will  help  us  to  decide  upon 
the  most  likely  ways  of  finding  an  answer  to  the  question  which 
the  symbol  of  integration  puts  to  us ;  namely,  W^hat  is  the 
fundio7i  whose  differential  coefficient  is  the  given  exj^ression?  To 
answer  this  question  is  in  very  many  cases  impossible ;  but 
practice  soon  makes  it  easy  to  recognise  the  simple  cases  which 
can  be  treated  with  success. 


*  In  Chapter  VIII.  we  shall  learn  that  there  is  another  way  of  looking  at 
integration  :  that,  in  fact,  integration  is  simply  a  summation,  or  more  exactly 

an  integral  is  the  limit  of  a  sum  :  and  that  the  symbol   /   of  integration  stands 
for  a  capital  S,  denoting  that  a  sum  is  being  taken. 


THE  INTEGEAL   CALCULUS— INTEGRATION       95 
The  follc^wiiig  is  the  table  of  Standard  Forms  : — 

(ii-)  J-^  =  log  a;,  since  ^  (logic)  =  -,        {x>0)     ^' 

(iii.)   [e'^dx         =^6*^,     ' 


(iv.)   m'dx  =Trrr^' 


a 
\_ 

log  a"'' 


(v.)  lcosa;c?a;     =sina7, 

(vi.)  Isinict^a;     =  -cosa;, 

(vii.)  I  tan  xdx     =  log  (sec  x), 

(viii.)  Icosecajf/jc  =log(tan^j, 

(ix.)  Isec^ccf/.^     =tanc):;, 

(x.)  |cosec-ir^a;=  -cot.r, 

^'^'-^Ivra  ='"''1'''"  ("'"'i)  ('''^'''^ 

,  ..  ,    {    dx  1         ,  a;  1         ,x 

(xii.)    1-^ s       =-tan     -   or    — cot  ^ -, 

^  ja^  +  x^  a  a  a         *  a 

(xiii.)     ~.^ r,       =?rlog ,     {xr>a^) 

(The  logarithms  are  to  the  base  e,  and  the  angles  are  measured 
in  radians.) 

The  student  is  recommended  to  draiv  up  a  corresponding  table  for 
the  cases  where  mx  +  n  takes  the  place  of  x  in  this  list. 


96      THE  INTEGKAL  CALCULUS— INTEGRATION 
§49.    Two  General  Theorems. 

(i.)   l(cw)c?.T  =  cpc?a:, 

(ii. )   I  (m  4-  v)dx  =  I « f^.?:  +  \vdx, 

€  being  a  constant,  and  u,  v  functions  of  x. 

In  the  first  theorem,  the  left-hand  side  of  the  equation  asks  us 
a  question :  What  is  the  function  whose  differential  coefficient 
is  6'w? 

The  right-hand  side  tells  us  that  the  answer  to  this  question 

is  clw6/ic. 

We  need  only  verify  this  answer. 
To  do  so  we  differentiate  cxudx. 


We  have  -j-c\udx=^- c^- \udx  =  cu, 


since  differentiating  an  integral  simply  cancels  the  symbol  of 
integration  and  the  dx. 

It  follows  that  I  cw^cc  =  c  I  udx. 


I  cudx  =  cU 


In  the  second  theorem,  the  left-hand  side  of  the  equation  asks 
us  a  question  :  What  is  the  function  whose  differential  coefficient 
is  u  +  v'i 

The  right-hand  side  tells  us  that  the  answer  to  this  question  is 

prfa;-f  \vdx. 
We  need  only  verify  this  answer. 

To  do  so  we  differentiate  pc?a;-|-  lt;<;?a:. 

We  have      -i-(  L(^«-|- Uc?x  j  =  -y-lwri?a:-|-^|i;c?.'c 

=  u  +  v. 


It  folio ws  that  I  (tt  +  t;)c?«  =    w  c?a;  -h  U  dx. 


THE  INTEGRAL  CALCULUS— INTEGRATION       97 

Using  these  theorems  we  can  readily  integrate  any  ordinary 
algebraical  expression  of  the  form 

ftQx"  +  a^x""^  +  . . .  +  a^_^x  +  a". 
Also,  if  we  call  this  expression  /(a;),  we  can  integrate 

x-a 
We  have  only  to  divide  fix)  by  (a;  -  a),  and  integrate  each  term 
of  the  quotient.     If  there  is  a  remainder,  the  corresponding  term 

fdx 
,  or  log  {x  -  a). 
x  —  a 

If  there  are  several  factors,  the  work  in  Algebra  on  Partial 

Fractions  comes  to  our  aid.     If  the  numerator  is  of  the  same  or 

of  higher  degree  than  the  denominator,  it  must  be  divided  by 

the  denominator  before  finding  the  partial  fractions. 

Ex.  1.  \{ax'^  +  2hx  +  c)dx  =  a\x^dx  +  2b\xdx  +  c\\dx'f' 


i  {ax"^  +  26a;  +  c)dx  =  a    x^dx  +  2b    xdx  +  cj] 

ax^    ,   o 

=  —+hx^  +  ex. 


jdx  +  2J, 


\dx 

dx 


2x-\ 

:.r  +  log(2a;-l). 


3.  /•^=a(_L_  _L)<i, 

J  x'^-a^     J  2a  \x-a    x  +  aj 

_J_  r  dx       1    f 
2a J  x-a     2a J 

2a     ^\x  +  a)' 
4.  /sin2xdx=/ — ^^-^dx  =  - / dx-- / cos2xdx 


dx 

x  +  a 

=  ?i-lofir( ),     when    x>a. 

2a     ^\x  +  aj' 


x_sin2x 
2         4     ' 


K  r      9    J         ri  +  cos2x^       X 

5.  /cos2xdx=/ ~ dx  =  - 

J  J  2  2 


X    sin  X  cos  X 

2^         2        ' 


*  \l dx  is  usually  written  as  \dx. 

t  This  is  one  of  the  standard  forms. 
C.C.  G 


98      THE  INTEGRAL  CALCULUS-INTEGRATION 

6.    Fill  up  the  blanks  in  the  following  table  : — 


fix) 

2x  +  - 

X 

x^  +  x+l 
x' 

Zx^  +  2x  +  -^ 
x+l 

3x''  +  -J—r 

3-^+/l 

x+l 
x  +  2 

a;2  +  2x+l 

x  +  2 

fnx)dx 

Many  of  the  most  important  applications  of  the  Integral  Cal- 
culus involve  only  such  integrals  as  we  have  now  learned  to 
calculate.  The  student,  who  has  not  time  to  take  up  the  question 
of  integration  more  fully,  could  omit,  in  the  meantime,  the 
remaining  articles  of  this  chapter. 

§50.   Integration  by  Substitution. 

To  prove  that    f{x)  dx  =  I  f{x)-rrdt,  where  x  =  <f)  (t). 

This  important  result,  which  allows  us  to  change  an  integral 
with  regard  to  x  into  an  integral  in  terms  of  another  variable, 
may  be  deduced  at  once  from  the  rule  for  differentiating  a 
function  of  a  function. 

Let  y  =  I  f(x)  dx    and     x  =  cfi  (t). 

From  the  relation  between  x  and  t,  y  is  a  function  of  t. 

But  dyjjdo'.^ 

dt     dx  dt 
dy     r,  .dx       .  dy     ,.  , 


y- 


f/(^- 


.dx  J. 


by  the  definition  of  an  integral. 

The  expressions  under  the  sign  of  integration  are  supposed 
given  in  terms  of  t. 

This  result  may  be  written 

The  simple  rule  for  "  changing  the  variable  "  from  xtot\^\ 

Replace  dx  hy  -r-  dt,  and  by  means  of  the  equation  connecting  x  and 

^    at 

t,  express  f{x)  as  a  function  of  t. 


THE  INTEGEAL  CALCULUS— INTEGEATION      99 

The   advantages   of   this   method   will  be   evident   from   the 
following  examples : — 


Ex.   1.  I  {ax  +  b)''dx.      Vnt  ax +  b  =  u. 

.    c?rc     1 
du~  a 

f  C      \  I  f  w"+^  1 

Thus      /  [ax  +  by^dx  =    ii''-du  =  ~  /  u'^du  =  —, r-  =    ,  (ax  +  b)''+^ 

J  J       a  a  J  a{7i+l)     a{n  +  l)^ 

Similarly  2.  /  sin  {ax  +  b)dx  =  -  j  s[nudu= =  -  -  cos  {ax  +  6). 

3.  /    ,.  -.     Put  ax  =  u. 

J  sJa^x'^-b'^ 

dx_l 
du~  a' 

Thus        f_iL==f_J_irf^=i/'_i^ 


=  -log{u  +  'Ju^-b'^) 
=  ~  log  {ax  +  s^a^x'-b"^). 

I^^^dx.         Put  a;  =  e^ 
J      X 

.    dx 


Thus  [  1^-  dx  =  [^  e«  du  =  ju  du  =  i  «2 


=  |(logx)2 


5.  [ ^^.  Putcc  =  cos^. 

•/(1-cc) 


dx 

dx 


de=-''''^' 


Thus  ( ^^=  =  [-— J-— _  (  -  sin 

J  {l-x)Jl-x^    j  (l-cos^)sm6»' 


sin^ 


:COt2 

\-x 


100     THE  INTEGRAL  CALCULUS— INTEGRATION 

6.  Integrate  the  following  expressions  : — 
(a)  a;"-i(aa;"  +  6) ;     xs] a?  ■\- x^ -,     Ya^^' 

^^^  x^Ax  +  2'    0:^  +  2^+2'  P^^*^"^  ^*+^==^'- 

^""^  ax^  +  2bx  +  c''    ax^+2hx  +  c    P^^^^"g  ax  +  h  =  n.     (aOb^) 

/s\  1  ^  +  2 

^°'     I  o     A       ^?       /  o     ,        =>  putting  a;  +  2  =  w. 

slx^  +  ^x  +  5      \/a;2  +  4a;  +  5     ^  * 

(e)  ■-    ^       ^=:r  ;       ■  — — »   putting  aa; +  &=:?«.     {ac>h'^) 
\/ax^  +  2bx  +  c      slax^  +  2hx  +  c     ^         ^  '  ' 

(f)  sin^xcos^a; ;     r—. — -;     cotar,  putting  sina;  =  w. 

(t  ~T~  0  Sin  X 

iv)  -5 o — ,70  •  9  -  ;     — s — r— ij-  >  putting  tanx  =  tt. 

a^cos^ar  +  ft^sin^.c      cos^ajsm^a;     ^  ^ 

§  51.   Integration  by  Substitution — continued. 

Although  there  are  certain  general  principles  that  guide  us 
in  the  choice  of  a  suitable  substitution,  a  second  form  (B)  of 
the  theorem  of  §  50  will  often  suggest  what  the  transformation 
should  be.     We  have  seen  that 

\fm)]ji[m]dt  =  ^f(x)dx,  where  x  =  <f>{ty 

We  may  write  this  result  in  the  form 

(B)  [f[H^)]^^[Hx)]dx=^  [f{u)du,  where  u  =  <j>{x)* 

as  the  particular  symbol  we  employ  is  immaterial. 

Thus  in  the  case  of  the  examples  of  last  article  we  obtain  our 
results  immediately — 


.g,  (i. )   [{ax  +  hfdx  =  -  \iax  +  hf^  {ax  +  h)  dx 
=  -  I  u'^duj  where  ax  +  h  =  u, 


=  — ..^  {ax  +  6f +1. 

71+1     a       a{n-\-l) 


*  This  can  be  verified  by  starting  with 

\  f{u)du, 
and  putting  w=0(a;),  as  in  (A). 


THE  INTEGRAL  CALCULUS— INTEGEATION     101 

(ii. )    I  sin^ic  cos  xdx  =    sin^ic  —  (sin  x)  dx 

=  I  uHu,  where  ^\nx  =  u, 

1    •   3 
=  ^  sin^a;. 

=  -  I  — ,  where  i*  =  1  +  x^, 
5J  u 

=  i  log  (1+0^5). 
o 

(iv.)  j?^^dx  =  logf(x). 

In  this  way  it  is  easy  to  see  that 

since  the  integral  may  be  written  as 

*.6.  -    — ,  where  u  =  ax'^  +  2bx  +  c. 

2J  ?* 

Also         f  -^-4?; =  f , j:r^ r,  ^  (ax  +  b)  dx 

J  ax^  +  2bx  +  G    J  (ax  +  bf  +  ac-b^  dx^  ' 

=  I  -s 775^^5  where  u^ax-vb. 

ju^-hac-b^ 

and  this  is  one  of  the  standard  forms. 

It  follows  that  any  expression  of  the  form 

Ix  +  m 


ax^  +  2bx^-c 


102  THE  INTEGRAL  CALCULUS— mTEGRATION 

can  be  easily  integrated,  since  we  can  rewrite  the  numerator  as 

-p{ax-\-b)  +  q,  ^ 

where  P  =  -  ;  0  = • 

a  a 

If  higher  powers  of  x  occur  in  the  numerator,  we  must  first  of 
all  divide  out  by  the  denominator  till  we  obtain  a  remainder  of 
the  first  degree  or  a  constant.* 

The  expression    ,    „       ,=-  may  be  reduced  in  a  similar  way. 
^  s/ax^  +  2bx  +  c       ^  ^ 

Ex.    Integrate  the  following  expressions  : — 
(i  )  _J_  •  1  1  37+1  2a;  +  3 


x^     .     a;2-a:+l.         x-\       .        x^-\-x-\-\ 
(ii.) 


x^±V    a;2  +  a;  +  l'    x^-S.'c  +  G'     (x- l)(a;-2)' 
1.1  1  x+\  2x  +  S 


s/x^±4'    \/a%2±?,2'    V4a;2  +  4a;±3'    \/4a;2  +  4x±3'    \/5  +  4a;-x2' 

§  52.    Integration  by  Parts. 

The  second  important  method  in  integration  is  called  integra- 
tion by  parts,  and  can  be  used  only  when  the  function  to  be 
integrated  is  the  product  of  two  functions,  one  of  which  can  be 
expressed  as  a  differential  coefficient.  This  method  follows  at 
once  from  the  rule  for  the  differentiation  of  a  product.  * 

Q.  d  ,     .        dv       du 

U/Jj  U/Jj  UjJU 

we  have  uv=  \(uj^  +^;7~)  ^^'  V  *^®  definition  of  integration, 
^^/idx  +  ^vfUx,  by  §49. 
It  follows  that         |i*-^G?a;=  wi;  -  h 


dv  ,  ^  du  -. 

W^rdx. 
dx 


n        7 

This  result  will  only  be  of  use  if   \v-j-dx  can  be  more  easily 
-    -       Cdv^~  J  ^^ 

J    dx 


evaluated  than  \u-^dx. 


*When  the  factors  of  the  denominator  are   real,    the   method    of   Partial 
Fractions  should  be  employed. 


dx 


THE  INTEGRAL  CALCULUS— INTEGRATION     103 
For  example— 
(i.)   \x\ogxdx  =  A\ogx^{x^)dx 

=  ^(x'^  log flJ  -  I  x^  ^ (log X)  dx^ 

=  -f  aj^logo;-  Li 

V   21  ^' 

=  -^[xnogx-^ 

=  ^(2log.-l). 

(ii.)   \x^cosxdx=\x^j-(smx)dx 

=  x'^&inx-  I  sin  a;  -j-  (x'^)  dx 
=  ^2  sin  ic  -  2 1  sin  a:  a;  c^a; 

Xj-  (cos  a:)  c?« 
=  a;2  gin  /^  ^.  2  f  :^  cos  X  -  I  cos  x  j-  (x)  dx 
=  a;2  sin  a;  +  2  (x  cos  x  -  I  cos  x  dx 


=  X-  sin  x+2x  cos  a;  -  2  sin  a;. 

In  both  of  these  examples  this  artifice  allows  us  gradually  to 
reduce  the  integral  to  one  of  a  simpler  form. 

In  such  cases,  where  powers  of  x  are  associated  with  a 
trigonometrical,  exponential,  or  logarithmic  term,  it  is  of  great 
value.  "^ 

An  important  expression  which  can  be  integrated  by  this 
method  is  \/a^  -  x^. 

*  Cf.  p.  105;  Exs.  11,  12,  13,  14,  and  15. 


104     THE  INTEGRAL  CALCULUS— INTEGRATION 
We  have 

I  sja^  -  x^dx  =  I  sja^  -x^-^-x  dx 

X  -rj-sjw^  -x^dx 
dx 

=  Xs/aF-^  +  I  ■     dx 


=  xja^  -x^  -  I  - 


».2  _  ^'2 


dx 


sja^-x^ 

dx 


=  xsla^  -  x^  -  f  v/a2  -  x'^dx  +  a^  f-^ 
.-.  2  I  \/«2  -x^dx  =  xsja^  -x^  +  a^ sin " ^ 


.-.  fx/a^-x^dx^^v/a^-x^+^sin-i-. 

The  expressions  \/x^ -  a^  and  JaF+i^  can  be  integrated  in  the 
same  way. 

Ex.  Integrate  the  following  expressions  : — 

x^  log  X  ;     a;V  ;     x  tan~^x  ;     x^  sin  ax  ;    sjd-x^  ;    Jx^  -  9. 

EXAMPLES   ON  CHAPTER  VII 

1.  Integrate  the  following  expressions  : — 

..  .  ....  1  1  +  a;      07  +  2 

(1.)  {x-aY,    -===.,     -— ,     — — . 

si  ax  +  h       six       x  +  6 

(ii  )  1  2a;- 1  a:^  x^ 

a;(l-a;)'     x^-Zx  +  2'     a;2  +  a;+l'     x'^-x+l' 
(iii.)  1  2a; -1  a:+l 


slx{\-x)      v/a;2-3a;  +  2      sIW+xTl 

2.  Integrate  the  following  expressions  by  parts  : — 

8in~^a;,     a;2tan~^a;,     a;^  sin  4a;,     a;^  cos  3a;,     a:*"  log  a;,     aj^e"* 

3.  Prove  that  _l_=:_i £Z^ 

■x^  +  l     3(a;  +  l)     3(x2-a;+l) 

and  hence  integrate  the  expression. 

4.  Prove  that 

1  111 


(a;+l)(a;-l)''^     2(a;-l)2     4{a;-l)     4(a;+l) 
and  hence  integrate  the  expression. 


THE  INTEGRAL  CALCULUS— INTEGRATION     105 

5.  Prove  that — —  = -, 

{x-2){x-S)     x-S    x-2' 

and  hence  integrate  the  expression. 

6.  Integrate  the  expressions  a;\''l  +a;  and  — — =  by  putting  x+l=u\ 

^\l  +  x 

7.  Prove  that         / t-         —  — -tan~^"  (put  a;  =  sin^). 

J  {i+x^)-Jl-x'    sl2  sll-x'' 

8.  Integrate  the  following  trigonometrical  expressions  : — 
111  1  sin  a; 

sin  ^'     sin  {6  +  a)'     sin  d  +  cos  ^'     cos^dsja^ tsm^f+h^'     cos"^a;(4 tan^a;  +  3)* 

9.  Show  that,  when  a^  >  b"^, 

/ r = -tan  M  A/ rtan-:r  . 

Putct  +  6cosa:  into  the  form  (a  +  6)cos^^  +  (a-6)sin2^. 

Also  integrate  the  expressions 

1111 


5  ±  4  cos  X      4  ±  5  cos  x      3  ±  2  sin  x      2  ±  3  sin  x 

10.    Prove,  by  integration  by  parts,  that 

,.  ,    /*  „^        ,     ,       6  sin  6x  +  a  cos  6a;  , 
(i. )    /  e**^  cos  hxdx  = ^.^  ^  ^^.^ e* 


(ii. )    /  e"^  sin  hxdx  _-  "'""^^     ^^v.^^.^.^^^ 


62 

a  sin  bx  —  b  cos  6a; 

a^  +  b-' 


11.    Prove,  by  integration  by  parts,  that 

cos  6  sin"-^^     n-1 


f  •   „n^n         cos^sni"-!^     71-1  r  . 

I  ain"ddd= + /sin"-^ 

J  n  n    J 


■^ddd, 
n  n    J 

and  hence  show  that 


r  .    . -  ,-         sinS^cos^     3   .    ^        ,3^ 
/  sin^^fZ^^ J rrsm^cos^-f^^. 

12.  Prove,  by  integration  by  parts,  that 

r         -,^     sin^cos"-!^     (71-1)  C  ^^  ,^ 

J  '^  n     J  ' 

and  thus  obtain  the  value  of  /  coa^ddd  and    /  cos^ddd. 

13.  Prove  that         \x^e^dx:=x'^e^-n\x^^-^e^dx, 

and_explain  how  this  result  may  be  used  in  evaluating  such  integrals  as- 
ixh^dx,    jxh-^dx,  etc. 


106     THE  INTEGRAL   CALCULUS— INTEGRATION 

14.  Prove  that  x''-'^  {log  x)'^dx=  I  y"'e''^di/, 

where   x  =  e^,   and   explain   how   this  result   may   be   used  in  evaluating 
integrals  such  as  r  r 

I  a;2(log  x)^dx,      /  a;-2(log  x)'^dx. 

15.  Prove  that 

r       .  ,  a;"  n   C       -. 

I  x^'  sin  mx dx= cos  mx  ^ —  /  x'"~  ^  cos  mx dx 

J  *'i  ^  j 

x'^  n     ^   .   .  n.n-  ]   C 

= cos  771X  H — r.a:"--'  Sin  mx „—  /  x^-^sm  mxdx, 

m  m^  m^      J 

and  show  how  this  may  be  used  in  evaluating  such  integrals.  *     Obtain  a 
corresponding  result  in  the  case  of 

/  x^ cos,  mxdx. 


*  Examples  3,  4,  5  are  cases  of  the  use  of  the  method  of  Partial  Fractions 
in  the  integration  of  algebraic  functions ;  11-15,  of  the  method  of  Successive 
Jteduction.     Cf.  Lamb's  Infinitesimal  Calculus,  §§  80,  81. 


CHAPTER  VIII 

THE   DEFINITE   INTEGRAL   AND   ITS   APPLICATIONS 

§  53.    Introductory. 

In  the  last  chapter  we  have  considered  the  process  of  inte- 
gration as  the  means  of  answering  the  question :  What  is  the 
function  whose  differential  coefficient  is  a  given  function  ? ,  As 
we  have  already  mentioned,  there  is  another  and  a  more  im- 
portant way  of  regarding  the  subject,  in  which  integration--^, 
appears  as  an  operation  of  summation,  or  of  finding  the  limit  ^ 
of  the  sum  of  a  number  of  terms.  We  shall  examine  inte- 
gration from  this  standpoint  in  the  following  sections. 

§  54.   Areas  of  Curves.    The  Definite  Integral  as  an  Expression 
for  the  Area. 

Let  y  =f{x)  be  the  equation  of  an  ordinary  continuous  curve, 
and  let  us  consider  the 
area  enclosed  between 
the  curve,  the  ordinates 
at  Po(xo,  ^o)  and  P(«,  y), 
and  the  axis  of  x. 

We  assume,  to  begin 
with,  that  PqP  is  above 
that  axis. 

This  area  is  obviously 
a  function  of  x,  since  to 
every  position  of  P  there 
corresponds  a  value  of  the 
area. 

Let  A  stand  for  the 
area   P^MoMP;   A-i-SA  for  the   area  P^M^NQ;  and  let  Q  be 


108  THE  DEFINITE  INTEGRAL 

the  point  {x  +  8x,  y  +  8y).  Then  if  the  slope  is  positive  from  P 
to  the  point  Q,  we  see  by  considering  the  inner  and  outer 
rectangles  at  P  and  the  element  of  area  there,  that 

y8x<8A<{y  +  8y)  8x. 

oA 

It  follows  that  y<-^  <y  +  ^' 

If  the  slope  is  negative,  the  signs  are  reversed. 

Hence  in  each  case,  when  we  let  8x  approach  its  limit  zero, 

we  have  , . 

dA  J.,  . 

Thus  if  we  write  'P{x)  for   l/(a;)c?a:,  and  if  C  stands  for  an 
arbitrary  constant,  we  have       ^ 

A  =  F(cc)  +  C. 

Also,  since  A  vanishes  when  a;  =  ccQ,  C=  -^{x^^. 

.-.  A  =  F(x)-F(ic„). 

This  expression  F(a;)  -'F{x^ 

is  an  important  one,  and  the  symbol 

f{x)dx 
is  used  to  denote  it. 

I    f(x)dx  is  called  the  definite  integral  of  f(x)  with  regard  to  x 

Jxq 

bet2veen  the  limits  x^  and  x-^ ,  and  its  value  is  obtained  by  subtracting 

the  value  of  the  indefinite  integral  \f{x)dx  for  x  =  Xq  from  that  far 
x  =  x^.  •' 

With  this  notation  the  area  of  the  curve  y  =f{x)  included  between 
the  ordinates  at  {Xq,  y^)  and  {x^,  y^),  the  axis  of  x  and  the  curve  is 


fXi 
f{x)dx. 
Xq 


It  can  be  shown  by  a  similar  argument,  or  otherwise,  that  if 
the  curve  cuts  the  axis  between  the  limits  x^  and  iCj,  the  definite 
integral  gives  the  algebraical  sum  of  the  areas,  those  above  the 
axis  of  x  being  taken  positive,  those  below  the  axis  negative. 


AND  ITS  APPLICATIONS  109 


Ex.  1.    To  find  the  area  of  the  sine  curve 
y  =  asinbx 

from  x  =  0  to  x  =  ^' 

2o 

The  required  area  =  I     a  sin  bx  dx 

Jo 


\ 


r  «    7. 1'' 

=       -  r  COS  OX  A 


~b 
This  notation   [F(a;)]^^  for  F{x-y)  -¥{xq)  is  useful  in  evaluating  Definite 
Integrals. 

2.  To  find  the  area  of  the  curve 

y=^as\n'^bx 

from  x  =  0  to  x  =  -r' 

b 

The  required  area  =  /    asm^bxdx 
Jo 

=  n  \    ( 1  ~  cos  2bx)  dx 

_  a  r       sin  2bx~\  * 

-2r"^2r"Jo 

_aTr 
~2b' 

3.  To  find  the  area  of  the  part  of  the  parobola  y^  =  4ax  cut  off  by  the 
lines  x  =  Xq  and  x  =  x-^. 


Here  the  required  area  =  2  /     \/4a:c  dx 

Jxq 

rxi       

—  4\/a  /     \lxdx 
Jh 

,-r2  3-1^1 


It   follows   that  the  area  cut  off  by  any  ordinate   P'NP  is  ^  of  the 
rectangle  upon  PP'  as  base,  with  AN  for  its  altitude. 

4.    To  find  the  area  of  a  circle  of  radius  a. 
Let  the  equation  of  the  circle  be 

x^-\-y"  =  a?. 

Then  the  required  area  =  4  /    sla^  -  x^  dx. 

Jo 
This  integral  can  be  obtained  by  substituting 

a;  =  asin^.     [Cf.  p.  115.] 


no 


THE  DEFINITE  INTEGRAL 


Or  we  may  quote  the  result  obtained  above  [§  52] : 

X     I Cb^  X 

\'a^  -x'^dx  =  n  va^  -  -v^  +  "2"  sin-i  - • 
Using  this  result,  the  area  of  the  circle  becomes 

2    x\la^-x^  +  a^sm-^-      : 
L  "Jo 


5.  Prove  that  the  area  of  the  ellipse  -.,  +  7 .,  =  1  is  irah. 

6.  Prove  that  the  area  between  the  hyperbola 

xy  =  c\ 
the  axis  of  x,  and  the  ordinates  at  {Xq,  y^),  [x^,  y{)  is 

^x_, 

,  Xn 

when  Xq  and  x-^  are  both  positive. 

7.  Prove  the  following  : — 

•^  dx 


M^ 


(i. 

(ii. 
(iii. 


(iv. 
(V. 
(vi. 


(ii. 
(iii. 

(iv. 

(V.) 


/. 


P   dx  r-  P 


sin^o:  dx  —  -=  I     cos^aj  dx. 
4     Jo 


dx 


p  dx  _  TT  _  p dx 

Jq  a^ sin^x  +  b^ cos^x~ 2ab~ J q  a^co8^x  + 

I    sin~'^xdx=  I     dcosddd—---!. 

Jo  Jo  ^ 

Jo  sla-x 

r^  dx  _  TT 

J\  xs/x"  -  i     3 
8.  Prove  that  when  w  and  n  are  positive  integers 

(i 


Jq  2n    .     2n-2...4.2  2     Jq 

sm''^dcos'"ddd  = /    sin'™-2^cos'*^c?^. 

Jo  m  +  nJo 

l\in^eGosHde  =  y^- 

IT  V 

r sin6 d coss ddd=  ,^'^'\^  I  '  cos^ 6 
Jo  14.  12.  10  Jo 


5t 
212' 


AND  ITS   APPLICATIONS 


111 


In  cases  where  integration  is  not  possible  there  are  various 
approximate  methods  of  finding  the  area.  The  expressions  for 
the  area  of  a  trapezium  or  a  portion  of  a  parabola  give  the 
trapezoidal  and  parabolic  rules, ^  and  we  shall  see  more  fully  in 
§§  55-56  how  the  inner  and  outer  rectangles  may  be  applied. 
The  value  of  a  definite  integral  may  also  be  obtained  by 
mechanical  means  by  the  use  of  different  instruments,  of  which 
the  planimeters  are  perhaps  the  best  known. 

Ex.  Evaluate  the  following  integrals  by  the  trapezoidal  method,  i.e. 
find  the  sura  of  the  inscribed  trapeziums  corresponding  to  the  divisions 
named  : — 

rl2 

(i.)    /     x^dx,  dividing  the  interval  into  11  equal  parts,  and  compare 

with  the  result  of  integration. 

Answers,  577^ ;  575§. 

(ii).    I      oosxdx,   by   dividing    the   interval   into  6  equal   parts,   and 
Jin" 
compare  as  above. 

Answers,   '0148;    -0149. 


55. 


The  Definite  Integral  as  the  Limit  of  a  Sum. 

In  the  last  article  we  have  shown  that  the  symbol  I  /W^^ 
represents  the  area  be-  ^  -^^ 

tween  the  curve 

the  axis  of  x,  and  the 
bounding  ordinates. 
We  shall  now  obtain 
an  expression  for  this 
area  as  the  limit  of  a 
sum,  and  thus  see  in 
what  way  the  process 
of  integration  may  be 
viewed  as  a  summa- 
tion. 

Let    PqPj    be    any 
portion  of  the  curve  in  which  the  slope  remains  positive. 


Fig.  23. 


*  Cf.  Lamb's  Calculus,  §  112 ;   Osgood's  Calculus,  p.  406 ;   Gibson's  Calculus, 
p.  329. 


112  THE  DEFINITE  INTEGEAL 

Divide  the  interval  M^Mj  into  n  equal  parts  Sx,  so  that 
n^x  =  x-^  -Xq  : 
erect  the  ordinates  m-^p-^,  m^p^,  etc.;  and  construct  inner  and 
outer  rectangles  as  in  Fig.  23. 

Then  the  difference  between  the  sum  of  these  outer  rectangles 
and  the  sum  of  the  inner  rectangles  is  («/i-«/o)^^5  ^^^  ^^i^  '^^y 
be  made  as  small  as  we  please  by  increasing  the  number  of 
intervals  and  decreasing  their  size. 

Also  the  area  of  the  curve  lies  between  these  two  sums. 
Therefore  this  area  is  the  limit  of  either  sum  as  &  approaches 
zero. 

Now  the  sum  of  the  inner  set  of  rectangles 

=  [/(^o)  ^^  +/(''^o  ^^^)^x'^  -'■  +f{^Q  +  n-l  .  8x)  Sx] 

=  2    f{xQ  +  r8x)8x. 

But  the  area  is  [F  (x-^)  -  F  (Xq)]   where  F(x)=\f{x)dx,  and   we 

f(x)dx. 

Xo 

tX]  r=n-l 

f{x)  dx  =  'Lt  2     /{Xq  +  r  8x)  8x 

Xo  Sx->0         r  =  0 

nSx=(xi-Xo) 
xi 

=  Lt     ^f(x)8x,  written  shortly. 

Sx  -^0    Xq 

It  is  easy  to  remove  the  restriction  placed  upon  f{x)  that  the 
slope  of  the  curve  should  be  positive  from  P^  to  Pj ;  and  to  show 
that  this  result  holds  for  any  ordinary  continuous  curve  whether 
it  ascends  or  descends,  and  is  above  or  below  the  axis  in  the 
interval  Xq  to  x-^. 

It  is  only  necessary  to  point  out  that  in  the  case  of  such  a 
portion  of  the  curve  y=f{x)  as  is  given  in  Fig.  24,  the  area  of  the 
portion  of  the  curve  marked  II  will  appear  as  a  negative  area, 

and,  if  I  f{x)dx  ^  F  (x), 

^f{x)dx,    or    [F(5)-F(a)], 


is  equal  to  (I)  -  (II)  +  (III). 

The  great  importance  of  the  Integral  Calculus  depends  upon 
the   fact   that    many  geometrical  and    physical  quantities   {e.g. 


AND  ITS  APPLICATIONS 


113 


volumes  and  surfaces  of  solids,  centres  of  gravity  and  pressure, 
total  pressure,  radius  of  gyration,  etc.)  may  be  expressed  in 
terms  of  the  limits  of  certain  sums.  The  problem  of  obtaining 
these  quantities  is  thus  reduced  to  a  question  of  integration. 


1  1 

i  i  l\ 

M 

i=b 

x=cx                  0 

\!      1 

w 

Fig.  24. 

We  have  already  remarked  that  the  symbol  of  integration   I 

really  stands  for  the  large  S  of  summation,  and  it  was  in  the 
attempts  to  calculate  areas  bounded  by  curves  that  the  Integral 
Calculus  was  discovered. 

It  is  also  possible  to  begin  the  study  of  integration  with 
the  definition  of  the  symbol 

'f{x)dx 

as  the  limit  of  a  sum,  and  to  develop  the  whole  subject  from 
that  definition."* 

§  56.  The  Evaluation  of  a  Definite  Integral  from  its  Definition 
as  the  Limit  of  a  Sum. 

It  is  instructive  to  see  how,  by  algebraical  methods,  the  values 
of  certain  definite  integrals  may  be  obtained  direct  from  this 
summation. 


c.c. 


*Cf.  Lamb's  Calculus,  §§90,  91. 
H 


114  THE  DEFINITE  INTEGRAL 

For  example,  in  the  case  of  the  parabola 

we  can  obtain  the  area,  or  the  Definite  Integral,  as  follows 

r=0 

=  &[V  +  (x^  +  Sx)^  +  (x„  +  2&)2  +  («„  +  (n  -  1)&)2] 

using  the  results  for 

-    .        1  +  2  +  3+... +(7i-l),     and     1^ +  2^  +  ... +{n-\y. 
Therefore,  since  TiSic  =  (ft'i  -  a;^)), 


r=n-l 


j^Sx(x,  +  rSx)^  =  x,^x,-x,)  +  x,{x,-x,y-(^l-lyl(x'^'--x,y(l-l^(-2-ly 


r=0 

r=n— 1 

Lt      2     8x(iro  +  r8a-)2 

fix->0       r^O 


X.'^-Xr, 


Xo 

Ex.    Prove  in  the  same  waj^  that 

"2jn 


f-^1  /y   3  _  7.  3 


and 


Jo  »w 

r*  1 

.'0  a 


f{x)dx. 

Xq 

The   following   properties   of   the   Definite   Integral   may  be 
deduced  from  either  of  the  definitions  of  this  symbol : —  ■ 


I. 


f{x)dx=-\   f(x)dx. 

JXq  JXi 


AND  ITS  APPLICATIONS  115 


11.    rf{x)dx=      {   f{x)dx+rf{x)dx. 

Jxq  Jxo  J  ^ 


III.  The  integral  of  an  even  function  between  the  limits  -  a  and 
+  a  =  twice  the  integral  of  the  function  between  0  and  a. 

2  „ 


fa  C^  2 

x^dx  =  2\  xHx  =  -^a- 


IV.  The  integral  of  an  odd  function  between  the  limits  -a  and 
+  a  is  zero. 

sin^Ode^O. 


x^dx  =  0,  si 

-a  J  -^ 

Similarly  fsin"*^  cos^'^-^'O  dB  =  0, 


m,  n  being  positive  integers. 

Y.  In  applying  the  method  of  the  "change  of  the  variable" 
to  the  evaluation  of  definite  integrals,  we  need  not  express  the 
result  in  terms  of  the  original  variable.  We  need  only  give 
the  new  variable  the  values  at  its  limits  which  correspond  to  the 
change  from  Xq  to  x-^  in  the  variable  x,*  care  being  taken  in  the 
case  of  a  many-valued  function  that  the  values  we  thus  allot  are 
those  which  correspond  to  the  given  change  in  x. 


E.g.  I  \/a 

Jo 

-n 


x^dx 
cos^ddS,     putting  x  =  a  sin  0, 
(1+ cos  2(9)^(9 


4>-"):=4'- 


*If  we  integrate   /    sinxdx  b}'  the  substitution  sinx  =  2/^  i*  appears  at  first 
Jo 
that  we  obtain  zero  instead  of  2  for  the  result.     It  is  not  hard  to  trace   the 
reason  for  this  discrepancy,  and  this  example  shows  that  in  the  use  of  this 
method  particular  care  has  to  be  taken. 


116 


THE  DEFINITE  INTEGRAL 


GEOMETRICAL  AND   PHYSICAL  APPLICATIONS   OF  INTEGRATION 

§  58.   Application  to  Fluid  Pressure. 

The  determination  of  the  pressure  of  a  liquid  upon  a  plane 
surface  shows  more  clearly  than  many  other  examples  the  real 
meaning  of  integration. 

Let  us  take  the  simplest  possible  case.  Suppose  we  have  a 
rectangular  trough,  resting  upon  a  horizontal  plane,  and  that  Ave 
fill  it  full  of  water.  There  is  a  certain  force  pressing  out  the 
ends  of  the  tank  due  to  the  weight  of  the  water.  We  shall  find 
the  amount  of  this  force  for  one  end :  in  other  words,  the  whole 
p-essure  of  the  water  upon  an  end  of  the  trough. 

The  whole  pressure  is  made  up  of  the  pressures  distributed 
over  the  surface  considered,  and  another  problem  is  to  find 
where  the  resultant  pressure  acts.  The  point  at  which  it  acts 
is  called  the  Centre  of  Pressure. 


Fig.  25. 

Let  the  rectangular  section  be  of  breadth  b  ft.  and  depth  a  ft. 
(Cf.  Fig.  25.)  We  take  the  line  AD,  which  lies  in  the  surface  of  the 
water,  as  the  axis  of  y,  and  the  vertical  line  AB  as  the  axis  of  x. 

We  learn  from  Physics  that  the  pressure  per  sq.  ft.  at  a  depth 
X  ft.  below  the  surface  of  the  water  is  wx  lbs.,  w  being  the 
weight  in  lbs.  of  a  cubic  ft.  of  water.  [1  cub.  ft.  of  water 
weighs  1000  oz.  or  62 J  lbs.] 

Suppose  that  the  rectangle  (cf.  Fig.  25)  is  divided  into  n  equal 
strips  by  lines  parallel  to  the  axis  of  ^,  the  thickness  of  each 
strip  being  8x. 


AND  ITS  APPLICATIONS  117 

Let  a-j,  x^^  x^,  ...x^^  be  the  distances  from  A  to  the  points 
where  the  lower  edges  of  the  strips  cut  the  axis  of  x. 
Let  8F^  =  the  pressure  on  the  ?"'  strip. 
Then  we  have      wx^  h8x>hV^  >0, 

wx^  b8x>  8P2  >  wx-^^  b  Sx, 


wx^^  b8x>  SP„  >  wx„  _ib8x, 

since  the  actual  pressure  on  each  strip  is  greater  than  what  we 
would  obtain  if  we  were  to  take  the  pressure  as  uniform  over  it, 
and  equal  to  that  at  its  upper  edge  :  also  it  is  less  than  what  we 
would  obtain  if  we  were  to  take  the  pressure  as  uniform  over  it, 
and  equal  to  that  at  its  lower  edge. 

Now  the  difference  between  the  sum  of  the  numbers  in  the 
first  column  and  those  in  the  third  is  equal  to 

wx^b8x    or     w8xx  the  area  of  the  rectangle. 

It  follows  that  when  Sic^-O,  n8x  remaining  always  equal  to  a, 
the  two  sums  become  equal  in  the  limit. 

But  the  actual  pressure,  » 

8Pi  +  SP,+  ...+8P„, 

lies  between  these  two  sums. 

It  follows  that  this  pressure,  which  we  shall  denote  by  P,  is 
equal  to  the  limit  whei^  8x->0  of  either  sum. 

In  other  words, 


P  =   Lt     2  mc,. .  &5a;  =  I   wx.  bdx  =  \ww^b. 

6a;^.0  r  =  0  J 

It  will  be  noticed  that  the  pressure  is  equal  to  the  area  of  the 
surface  immersed  multiplied  by  the  pressure  at  its  Centre  of  Gravity, 
a  theorem  which  can  be  shown  to  be  true  in  general. 

We  take  another  example,  where  the  section  is  not  rectangular. 
Let  the  section  of  the  trough  be  a  semi-circle,  the  diameter  lying 
in  the  surface  of  the  water,  its  length  being  2a. 

In  this  case  it  is  convenient  to  take  the  origin  at  the  centre, 
and  the  axes  of  x  and  y  vertical  and  horizontal. 


118 


THE  DEFINITE  INTEGRAL 


As  before,  we  divide  the  section  into  strips  parallel  to  the  axis 
of  y,  the  breadth  of  each  strip  being  Sx. 

Let  QNQ'  be  the  upper  edge  of  one  of  these  strips.  Let 
ON  =  a;  and  QN  =  y.     [Cf.  Fig.  26.] 


Q  i^-y) 


Let  6P  =  the  pressure  on  this  strip. 
Then  we  have 

1w {x  +  8x) y8x>8'P  >  2ivx (y  +  Sy)  Sx. 

If  we  were  to  add  the  terms  in  the  first  and  third  columns 
obtained  in  this  'way,  we  would  find  that  the  limit  of  each  sum 
would  be  the  same,  when  8x-^0,  n8x  remaining  equal  to  a. 

Also  the  pressure  P  lies  between  the  two  sums. 

It  follows  that  P  is  equal  to  the  limit  of  either  sum,  and  in 
finding  this  limit  we  can  omit  the  terms  involving  SxSy.  The 
sum  of  these  terms  vanishes  in  the  limit. 


Thus 


P  =  2i^  I   xydx,  where  y  =  Ja^  -  x'^, 
Jo 

=  2w\  xja^-x'^dx 
=  2w\_-l{a^-x^~fX 

When  we  have  obtained  the  position  of  the  c.G.  of  a  semi- 
circle (cf.  p.  124),  we  shall  see  that  the  above  answer  agrees 
with  the  general  theorem  to  which  we  have  referred. 


AND  ITS  APPLICATIONS         .  119 

These  results  can  easily  be  extended,  and  a  general  formula 
obtained.  However  the  student  is  advised,  at  this  stage,  to 
work  out  such  examples  from  first  principles.  When  he  has 
grasped  the  meaning  of  the  argument  used  in  the  above  dis- 
cussion, it  is  unnecessary  to  write  down  the  inequalities  on 
which  it  depends  in  full.  For  example,  in  the  case  of  the  semi- 
circle, it  would  be  sufficient  to  say  that 

6P  =  2wx  .  y^x,  to  the  first  order ; 

so  that  on  integrating  over  the  semi-circle 

(a 
xydx: 
0 

Ex.  1.  A  water  main  6  ft.  in  diameter  is  just  full  of  water.  Show  that 
the  pressure  on  the  gate  that  closes  the  main  is  over  2|  tons. 

2.  A  vertical  masonry  dam  is  in  the  form  of  a  rectangle  200  ft.  long  at 
the  surface  of  the  water,  and  50  ft.  deep.  Show  that  when  full  it  has  to 
withstand  a  pressure  of  nearly  7000  tons. 

3.  The  bank  of  a  reservoir  is  inclined  at  an  angle  of  60°  to  the 
horizontal.  If  the  depth  of  the  water  is  30  ft.,  show  that  the  normal 
pressure  on  the  section  100  ft.  long  is  over  1400  tons. 

§  59.   Application  to  Areas  in  Polar  Co-ordinates. 

When  the  equation  of  the  curve  is  given  in  polar  co-ordinates, 
the  area  of  the  sector  bounded  by  ^  =  ^o  ^^^^  B  =  6^  can  be 
shown  to  be 

Lt       2  (■.rm 
se->o   e=eo\-^ 

with  the  same  notation  as  before.     Hence  if  the  curve  is  r=f{d), 
the  sectorial  area  is 


l[\f{d)fdd. 


Polar  co-ordinates  are  often  used  in  finding  the  area  of  a  loop 
of  a  curve. 

For  example,  the  lemniscate 

r^  —  a^  cos  26 


has  a  loop  between  6=  -  -  and  6  =  -r' 
,  4  4 


120  THE  DEFINITE   INTEGRAL 

1  r^ 

The  area  of  this  loop  =  -      ^  r\id 


cos  26 dd. 


.'.  the  area  of  the  loop  =  612^008  2(9^(9  (Cf.  §  57,  III.) 

-<^t   ■ 

_a2 
~  2  * 
Similarly,  in  the  Folium  of  Descartes,  whose  equation  is 
x^-\-y^  —  Saxy, 
there  is  a  loop  in  the  first  quadrant. 

Using  polar  co-ordinates  we  find  that  the  area  of  the  loop 


I!. 


Hd 


I  pf3acos^sin(9)2^^ 
cos^^  +  sin^^  I 


I     cos2^sin2^     ^^ 


IT 

9     f^ 

=  2^Jj,(cos3(9  +  sin3(9)2 

=  ^'(-1^73! 
~  2  * 

o 
Ex.    Prove  that  the  area  of  the  cardioide  r  =  a{l  -cos^)  is  ^^ra^. 

§  60.   Application  to  Lengths  of  Curves. 

The  length  of  an  arc  PqPj  of  the  curve  y=f(x)  may  be 
regarded  as  the  limit  of  the  sum  of  the  different  chords  inta 
which  PqPj  is  divided  by  the  ordinates  at  m^,  Wg,  ...  (cf.  Fig.  23)., 


AI^D   ITS   APPLICATIONS  121 

Hence 

arcPoPi  =  Lt       /'   'J{W+"{^W 

Sx  ->  0    X  =  Xo 

5.c->0    x=x,      >  VW 


-i:v-(i)"-f::v 


'*||)"*. 


since 


Jl  +  fJl-\  will  differ  from   Jl  +  (-l\  by  a   very   small 

quantity  when  8x  is  very  small,  and  the  sum  of  these  differences 
multiplied  by  8x  will  vanish  in  the  limit. 

If  polar  co-ordinates  are  used,  we  obtain  the  two  expressions 


since  the  chord  is  \/{8ry^  +  {r86)'^  to  the  first  order. 

Owing  to  the  presence  of  the  radical  sign  under  the  sign  of 
integration,  the  problem  of  finding  the  length  of  the  curve  has 
been  solved  in  only  a  limited  number  of  cases. 

Ex.  1.    Prove  that  the  length  of  the  arc  of  the  parabola  7/^  —  4ax  from 
the  vertex  to  the  end  of  the  latus  rectum  is  equal  to  a[>y2  +  log(\/2  + 1)]. 
2.   Prove  that  the  length  of  the  cardioide  r  =  a{l  -  cos^)  is  8a. 

§  61.   The  Volume  of  a  Solid  of  Revolution. 

Let  the  solid  be  formed  by  the  revolution  of  the  curve 

about  the  axis  of  x. 

We  wish  to  obtain  the  volume  contained  between  the  planes 
x  =  X(^  and  x  =  Xy 

We  suppose  the  interval  Xq  to  x^  divided  up  into  n  equal  parts 
8x,  as  in  §  55 ;  and  we  take  the  sections  of  the  solid  by  the 
planes  perpendicular  to  the  axis  at  these  points.* 

*  We  might  also  proceed  as  follows  : — 

We  have  Try^5x^dV^'ir(y+  5yf  Sx, 

V  standing  for  the  volume  up  to  the  section  considered,  and  SFfor  the  increment 
of  volume. 

Thu,  |^=.y^. 

The  rest  of  the  argument  presents  no  diflaculty. 


122  THE  DEFINITE  INTEGRAL 

If  we  let  inner  and  outer  discs  take  the  place  of  the  inner 
and  outer  rectangles  of  our  former  argument,  it  readily  follows 
that  the  required  volume  is  given  by 

r=x-l 

Lt      2    TTij^^Sx, 

'/I  Sx=Xi  —  Xo 

where  y^  =/(^o  +  ^  ^^)- 

Thus  the  volume  =  tt  I    y^dx  =  it  I    \^f{x)Ydx. 

JXa  JXo 

When  the  axis  of  y  is  the  axis  of  revolution,  the  area  of  the 
section  is  irx'^  instead  of  iry'^,  and  the  volume  becomes 


pi 
•  I   xHy. 

JVn 


Ex.  1.  The  portion  of  the  parabola  i/^  =  4£ix  from  the  vertex  to  the 
point  Vix,  y)  revolves  about  Ox.  Prove  that  the  volume  of  the  cup  we 
thus  obtain  is  lairx^. 

2.  Obtain  the  volume  of  a  sphere  by  considering  the  rotation  of  the 
semicircle  x^  +  y'^  =  a'^  about  Ox. 

3-  Find  the  volume  (i.)  of  a  right  circular  cone,  and  (ii.)  of  a  cone  in 
which  the  base  is  any  plane  figure  of  area  A,  and  the  perpendicular  from 
the  vertex  upon  the  base  is  h. 

4.    Prove  that  the  volume  of  a  spherical  cap  of  height  li  is  irh^  ( *'  -  o 
where  r  is  the  radius  of  the  sphere.  ^ 

§  62.   The  Surface  of  a  Solid  of  Eevolution. 

It  is  easy  to  show  that  the  surface  of  a  right  circular  cone 
whose  vertical  angle  is  2a  and  whose  generators  are  of  length  I 

is  ttP  sin  a.  We  can  deduce  from 
this  that  the  surface  of  the  slice 
of  a  cone  obtained  by  revolving 
a  line  PQ  about  Ox  is  equal  to 

27r .  PQ .  NR, 
where   NR  is  the   ordinate   from 
X  the  middle  point  of  PQ. 


O  N 

Suppose  then  that  an  arc  PqP^ 

Fig.  27.  c     ^  j"/   \ 

01  the  curve  y=j{x)  rotates  about 
Ox.  The  area  of  the  surface  generated  by  P^^P^  is  the  limiting 
value  of  the  sum  of  the  areas  of  the  surfaces  generated  by  the 
•chords  into  which  we  suppose  this  arc  divided.  Thus  the  area  of 
the  surface  generated  by  P^Pj 


AND  ITS  APPLICATIONS  123 


Sx-^0    x=Xo         \          -^       / 

M&^ 

=  27r j  V  ^1  +  {J^  dx,  where  y=f{x). 

yds,  by  changing  the  variable  from 

X  to  5,  where  s  is  the  length  of  the  arc  from  a  fixed  point  to  the 
point  (x,  y). 

When  the  axis  of   revolution  is  the  axis  of  y,  we  obtain  in 


the  same  way  the  expression  27r  I   xd^^. 

J  So 


Ex.  1 .    Obtain  the  expression  for  the  surface  of  a  sphere  of  radius  a. 
Here  we  take  the  curve  y  =  s]d^  -  x^, 

and  the  surface  =  47r  I    -Ja^ -  x^  \l\  +  -^dx 

ra 

=  47ra  /    dx 

•0 

2.  Prove  that  the  area  of  the  portion  of  a  sphere  cut  off  by  two  parallel 
planes  is  equal  to  the  area  which  they  cut  off  from  the  circumscribing 
cylinder  whose  generators  are  perpendicular  to  these  planes. 

3.  Prove  that  the  area  of  the  surface  formed  by  rotating  the  circle  of 
radius  a,  whose  centre  is  distant  d  from  the  axis  of  x,  about  that  axis, 
is  4:Tr^ad. 

§  63.   The  Centre  of  Gravity  of  a  Solid  Body. 

If  a  number  of  particles  of  masses  w^,  m^,  ...  are  situated  at 
the  points  {x^,  y^,  z^),  ...,  their  C.G.  is  given  by 

2(m^)  2(m^  2(m^) 

""-  2K)'    ^-  2K)'         2K)- 

Now  we  may  suppose  any  solid  body  broken  up  into  small 
elements  of  mass.     Let  {x,  y,  z)  be  the  C.G.  of  the  element  Sm. 
Then  we  may  write  these  results  for  a  solid  body  in  the  form 
Lt    ^xSm.  Lt    ^ySm  Lt    ^z8ni 

Sm  -^0  -       Sm  ->  0 


y 


M        '     ^        '     M       '  M 

In  many  cases  we  can  transform  these  expressions  into  integrals 
which  we  can  evaluate  by  the  methods  already  employed,  though 
in  general  they  involve  integration  with  regard  to  more  than  one 
variable,  and  such  integrals  cannot  be  discussed  here. 


124 


THE  DEFINITE  INTEGRAL 


We  add  some  illustrative  examples  : — 

Ex.  1.    The  Centre  of  Gravity  of  a  Semi-circular  Plate. 
Take  the  boundary  of  the  plate  along  the  axis  of  y,  and  suppose  the 
semicircle  divided  b}'^  a  set  of  lines  parallel  to  that  axis  and  very  near  one 

another.     The  C.G.  of  each  of  these  strips 
PQ'  lies  on  the  axis  of  x,  and  therefore  the 
C.G.  of  the  semicircle  lies  on  Ox. 
We  thus  have 

Lt    Sx  8m 
—     Sm-*0 
X  = 


M 


I j    xydx 


x'dx 


— r,  /    xs/a^ 
ra-Jo 


and  y 


0. 


2.   The  Centre  of  Gravity  of  a  uniform  Solid  Hemisphere. 

Let  the  axis  of  x  be  the  radius  to  the  pole  of  the  hemisphere,  and  suppose 
the  solid  divided  up  into  thin  slices  b}^  a        y 
set  of  planes  perpendicular  to  this  axis. 

Then  the  C.G.  of  each  of  these  slices 
lies  on  this  axis,  and  therefore  the  C.G. 
of  the  hemisphere  does  so  also. 


Also  we  have 


TTxy^dx 


2         3 


x^dx 


=24^(<'1II-[t]. 

_  3  /I  n 


3 


AND  ITS  APPLICATIONS 


125 


3.    Prove  that  the  C.G.  of  any  cone  or  pyramid  upon  a  plane  base  is  one 
fourth  of  the  way  up  the  line  from  the  vertex  to  the  C.G.  of  the  base. 


4.    Prove  that  the  C.G.  of  the  upper  portion  of  the  ellipse  -2+^ 
4?; ' 


at  the  point  (  0,  — -  ) 


1  is 


§  64.   Moments  of  Inertia. 

The  moment  of  inertia,  I,  of  a  set  of  particles  m^,  m^,  ...  with 
respect  to  an  axis  from  which  they  are  distant  7\,  r^,  etc.,  is  the 
expression 

m^r-^^  +  m.f^^  +  ... , 

and  in  the  case  of  a  continuous  solid  body  we  may  express  this  as 

1=   Lt   2r26m. 

Sm  -^  0 

The  radius  of  gyration  k  is  defined  by  the  equation 

l  =  MkK 

In  many  cases  we  can  obtain  the  values  of  I  and  k'^  by  the 
use  of  the  methods  of  integration  we  have  been  discussing. 
We  add  some  illustrative  examples  : — 

Ex.  1.     To  find  the  radius  of  gyration  of  a  thin  rod  of  mass  M  and  length 
21,  about  an  axis  at  right  angles  to  the  rod  and  passing  through  its  centre. 
Here    1=    Lt    'Lx^bm 


Sm^-O 

=  p         X^'dx, 

i  2lp  =  M, 

=  2p  f  x-'dx 

-'0 

0 

P 

=g^ 

0 

MP 

"  S  ' 

Fig.  30. 

■■■  '^  =  1'^^- 

2.    To  find  the  moment  of  inertia  of  a  solid  circular  cylinder  about  its 
axis. 

We  have  I  =    Lt    Sr^Sm, 

where  8m  =  p/t  {tt  (r  +  dr)"^  -  7rr-} 

=  7rph{2r5r  +  {Srf}, 
p  being  the  volume  density,  and  h  the  height  of  the  cylinder. 


16 

THE 

DEFINITE  INTEGRAL 

Therefore 

l  =  irph\    r-2rdr 

But 

3.    Prove  that  the  radius  of  gjration  of  a  thin  circular  plate  of  radius  a 
about  a  diameter  as  axis  is  -  a^. 


EXAMPLES  ON  CHAPTER  VIII 

1.  Find  the  areas  bounded  bj' 
(i.)  y  =  sin2x,  x  =  0,  x  =  ^. 

(ii.)  y  =  e-^  sin  20;,  x  =  0,  x  =  ^- 

(iii.)  The  hyperbola  xy  =  a^f  x  =  Xi,  x  —  x^. 
(iv.)  y  =  v(^,  'x  =  0,  x  =  A. 
(v.)  y  =  2oi^,  the  axis  of  y,  and  the  lines  y  =  2  and  ?/  =  4. 

2.  Find  the  area  of  the  part  of  the  parabola  y^^x"^ -^x  +  2  cut  ofif  by  the 

a;  axis.     What  does   /  yrfx  here  represent? 

.'o 

3.  Trace  the  parabola  {y -x-'^)'^  =  x-\-y^  and  find  the  area  of  the  part 
of  the  curve  cut  off  by  the  lines  x=^0  and  x  =  4:\. 

4.  Find  the  areas  in  polar  co-ordinates  of 

(i.)    The  part  of  r  =  ad  included  between  ^  =  0  and  d  =  2ir. 
(ii.)  A  loop  of  each  of  the  curves  r  =  a  sin  2^,  a  sin  3^,  etc. 
(iii.)  A  loop  of  each  of  the  curves  r  =  a  cos 2^,  a  cos  3^,  etc. 
(iv.)  The  part  of  the  hyperbola  r^ sin  ^ cos  ^  =  a^  included  between  6  =  6-^^ 
and  0  =  02. 

(v.)  A  sector  of  the  ellipse  -^.,  +  10,  =  ^  and  of  the  hyperbola     a" 72  =  ^' 
the  centre  being  the  pole. 

(vi.)  Prove  that  the  area  between   the  two   parabolas   y'^  =  '^ax   awd. 

x^  =  \ay  is  — ^ — 

x^    y"^ 
(vii.)  Prove   that   the   area    between    the    two   ellipses      2+72  =  1    ^^^ 

^'    2/'     ,   •     .   7.       1^  " 

,^  +  ^=1  IS  4a&tan-^-. 


AND  ITS   APPLICATIONS  127 

5.  'By  substituting  a:'  =  acos^,  y  —  h  sin  6,  show  that  the  perimeter  of 
the  ellipse  of  semiaxes  a,  b  is  given  by  4a  l  s! \  -  e^  s\n^  d .  dd ,  and  deduce 
that  for  an  ellipse  of  small  eccentricity  the  perimeter  is  approximately 

6.  Find  the  lengths  of  the  following  curves  : — 

(i.)  The  equiangular  spiral  r  =  ae^^°^*  from  ^  =  0  to  d  —  lir. 
(ii.)  The  spiral  of  Archimedes  r  =  ad  from  ^  =  0  to  d  =  2ir. 

(iii. )  The  catenary  y  =  -  /  €«  +  e   «  ]  from  x  =  0  to  x  =  a. 

(iv. )  And  show  that  the  length  of  a  complete  undulation  of  the  curve 

y  =  osin- 
a 

is  equal  to  the  perimeter  of  an  ellipse  whose  axes  are  2\la^  +  h'^  and  2a. 

7.  Find  the  volumes  of  the  following  solids  : — 

(i.)  The  solid  formed  by  revolving  the  part  of  the  line  x-\-y=\  cut  off 
by  the  axes,  about  the  axis  of  x,  and  verify  your  result  by  finding  the 
volume  of  the  cone  in  the  usual  way. 

(ii.)  The  spheroid  formed  by  rotating  the  ellipse  9a:2  +  162/"=144  about 
the  axis  of  x. 

(iii. )  The  cup  formed  by  the  revolution  of  a  quadrant  of  a  circle  about 
the  tangent  at  the  end  of  one  of  its  bounding  radii. 

(iv. )  The  cup  of  height  h  formed  by  the  revolution  of  the  curve  a'^y  =  x'^ 
about  the  axis  of  y. 

(v.)  The  ring  formed  by  the  revolution  of  the  circle  {x -a)'  +  y^  =  h'^ 
about  the  axis  of  y. 

(vi.)  The  ellipsoid  ^,  +  ^  +  ^^=1. 

And  show  that  if  Sq,  Sj,  Sg  are  the  areas  of  three  parallel  sections  of  a 
sphere  at  equal  distances  a,  the  volume  included  between  Sq,  S.2  and  the 

spherical  boundary  is  -  (Sq  +  48^  +  Sg). 
o 

8.  The  ellipse  whose  eccentricity  is  e  rotates  about  its  major  axis. 
Prove  that  the  area  of  the  surface  of  the  prolate  spheroid  thus  formed  is 


rb(^h  +  l, 


9.  The  catenary  y  =  -le'^  +  e  "  j  rotates  about  the  axis  of  y  ;  prove  that 
the  area  of  the  surface  of  the  cup  formed  by  the  part  of  the  curve  from 
a;  =  0  to  a;  =  a  is  27ra"(  1  —  j. 


128  THE  DEFINITE  INTEGEAL 

10.  The  cardioide  ?^  =  a(l-cos^)  revolves  about  the  initial  line;  prove 

that  the  surface  of  the  solid  thus  formed  is  —  ira^. 

5 

11.  Find  the  C.G.  of  the  following  :  — 

(i. )  A  thin  straight  rod  of  length  I  in  which  the  density  varies  as  the 
distance  from  one  end. 

(ii.)  An  arc  of  a  circle  of  radius  a  which  subtends  an  angle  2a  at  the 
centre. 


1. 


(iv.)  A  circular  sector  as  in  (ii.). 

(v. )  The  segment  of  the  sector  of  (iv.)  bounded  by  the  arc  and  its  chord, 
(vi.)  A  thin  hemispherical  shell  of  radius  a. 

12.    Find  the  moments  of  inertia  of  each  of  the  following : — 
(i.)  A  thin  straight  rod,  about  an  axis  through  an  end,  perpendicular 
to  its  length. 

(ii. )  A  fine  circular  wire  of  radius  a,  about  a  diameter, 
(iii.)  A  circular  disc  of  radius  a,   about   an   axis   through   its   centre 
perpendicular  to  the  plane  of  the  disc. 

(iv.)  A  hollow  circular  cylinder  of  radii  a,  h,  and  height  h,  about  its  axis, 
(v. )  A  sphere  of  radius  a,  about  a  tangent  line. 

(vi.)  (a)  A  rectangle  whose  sides  are  2a  and  2o,  about  an  axis  through 
its  centre  in  its  plane  perpendicular  to  the  side  2a  ; 

(/3)  about  an  axis  through  its  centre  perpendicular  to  its  plane, 
(vii.)  An  ellipse  whose  axes  are  2a  and  26, 
(a)  about  the  major  axis  a  ; 
(/3)  about  the  minor  axis  h  ; 

(7)  about  an  axis  perpendicular  to  its  plane  through  the  centre, 
N.B. — The  case  of  the  circle  follows  on  putting  a  —  h. 
(viii.)  An  ellipsoid,  semiaxes  a,  b,  c,  about  the  axis  a. 
jS^.B. — For  the  sphere  a  =  b  =  c. 
(ix. )  A  right  solid  whose  sides  are  2a,  2b,  2c,  about  an  axis  through  its 
centre  perpendicular  to  the  plane  containing  the  sides  b  and  c. 

N.B. — Routh's  Rule  for  these  last  four  important  cases  can  be  easily 
remembered : — 

^sum  of  squares  of  perpendicular\ 
Moment  of  Inertia  about  an  axis\  _  \  semiaxes  ) 

of  symmetry  j  -'^^^^  3,  4,  or  5 

The  denominator  is  to  be  3,  4,  or  5  according  as  the  body  is  rectangular, 
elliptical,  or  ellipsoidal. 

Cf.  Routh's  Rigid  Dynamics,  vol.  i.  p.  6. 


APPENDIX 

Alternative  Proofs  for  the  Differentiation  of  x^^,  e^  and  log  x. 

I.    The  Differential  Coefficient  of  x'\ 
Let  y  =  x"'. 

Then  y-\-hy  =  {x^  hxf 


^x  ^x 

But  by  the  Binomial  Theorem,  when  ^  <  1, 

(1  ^hf=\  +nh-^  '^'!'~^  ¥+.... 
Therefore 

8y        V       X I  .'Ix^ / 

oa:  1.2 

provided  that  Sa;  is  so  small  that 

X 

*  The  fact  that  we  have  an  infinite  aeries  on  the  right  hand  sometimes  causes 
diflBculty  to  the  student,  as  he  imagines  that  what  he  calls  the  summing  of  the 
nfinite  number  of  small  terms  involving  5a;,  (Sa;)^,  etc.  ...  may  give  rise  to  a 
finite  sura.  The  answer  to  this  difficulty  in  general  is  to  be  found  in  a  true  view 
of  the  meaning  of  a  convergent  infinite  series,  but  in  the  series  here  referred  to 
we  are  able  to  say  what  the  possible  error  can  be  if  we  stop  after  a  certain 
number  of  terms.  We  thus  exclude  the  infinite  series  from  our  argument. 
C.C.  I 


130  APPENDIX 


Hence  Lt 

Sx- 

and  the  differential  coefficient  of  «"  is  wf^'K 


'.o\8xJ 


II.    To  differentiate  e*. 
Let  y  =  e"". 

Then  y-^^y  =  e^+*''- 

.-.  hy^e^+^^-e'^ 
=  e^(e««-  1) 


6^(1+^  +  ^+.. .-1 


&x        V       2!^   3!    *      ) 


Proceeding  to  the  limit,     j^  =  «*• 
Thus  |^(«')  =  a'. 

III.  To  differentiate  logx. 
Let  y  =  log  a^. 

Then  y  +  ^y  =  log  («  +  8x). 

Therefore  8y  =  log  (x  +  Sic)  -  log  x 

=iog(i.;) 

8a;     l/8a;\2     I/Sa^V  -t 


<1. 


8«     X     2x\  X 

Proceeding  to  the  limit,     -/  =  -. 

"  dx     X 

Thus  -i(loga;)  =  -. 


a;     2a;V^/     ^x\x) 


ANSWERS 


CHAPTER  I.      (p.  15) 

1.  (1.)  a;2  +  2/2=    -2—  ^"-^  *'"4a" 
(iii.)  a;-'  +  2xV  +  2/^  +  2a2(2/2_a:2)  +  a.4-c^=rO. 

2.  .T  +  42/-ll  =  0.  3.    (-If,   jl). 

4,    The  parallel  lines  through  O  are 

Sx-2y  =  0,   4x  +  y  =  0,    19a^+13y  =  0. 
The  perpendicular  lines  through  0  are 

2x  +  37/  =  0,   x-^y  =  0,    13a;- 19^  =  0. 
The  parallels  through  (2,  2)  are 

3x-2y  =  2,   ^x  +  y  =  lO,    I9x+13y  =  64. 
The  perpendiculars  through  (2,  2)  are 

2x  +  3y=l0,   x-4ty  +  Q  =  0,    I3x-I9y  +  12  =  0. 
6.   x  +  3y-1  =  0. 

6.  Ix+ly  -  36  =  0  is  the  bisector  of  the  acute  angle. 

x-y -12  =  0  is  the  bisector  of  the  obtuse  angle. 

7.  (i.)  (1,2),   (3,4),    (5,3).  (ii.)  |,    -3.   ^. 
(iii.)  The  internal  bisectors  are 

x-y+l-x  +  4y-7     x-y+l _  -a;-2y  +  ll     x-  4y  +  7 _ x  +  2y-  11 

~vi"~    ^n    '     n/2   ~      n/5     '     x/n   ~    \/5    * 

The  external  bisectors  are 
x-y+l_x-4:y  +  7     x-y+l  _x  +  2y -II     x-^y  +  7  _  -a;-2y  +  ll^ 
v/2  /v/r7      '        \/2     ~       >/5       '        \/r7      ~         \/5 

8.  If  the  points  (0,  0),  (2,  4),  ( -  6,  8)  be  called  A,  B,  C  respectively, 

(i.)  BCis  a;  +  2y-10  =  0,  (ii.)  tanA  =  2,   tanB  =  oo,   tanC^^* 

CA  is        4x  +  3y  =  0, 
ABis  2x-y=0. 

C.C.  I  2 


132  ANSWERS 

(iii.)  Median  through  A  is  y  +  Sx  =  0, 

Median  through  B  is  y  -  4  =  0, 

Median  through  C  is  6^:;  +  7y  -  20  =  0. 
(iv.)  The  perpendicular  from  A  on  BC  is  the  line  AB  ;  its  length  is 
2v/5. 
The  perpendicular  from  B  on  CA  is  the  line  Sx-4y  +  ll=0  ;  its 

length  is  4. 
The  perpendicular  from  C  on  AB  is  the  line  CB ;  its  length  is 

4\^. 

4 
(v.)  x  +  2i/  =  0,  (vi.)x=-K'   2/  =  4. 

4a: +  32^ -20  =  0,  "^ 

2a; -y +  20=0. 

^'"■'  [     3  +  x/5      '     3  +  N/5  )'   ^       '    ^'    I     2'*; 
9.    ^1(^2 -y3)  +  a^2(.y3-2/i)+a:3(2^i- 2/2)-  l®.    (i.)  6.     (ii.)  40. 

CHAPTER  II.     (p.  29) 

4.    y-9a:+16  =  0.  5.    x  =  -i   y  =  ^:x  =  l,   y= -\. 

6.    n-gi ;    -  g.  7.    27rrA  5r. 

8.    5V  =  4Tr25r,  50-27,  502-66.  9.    (a  +  2bt),    5l  =  {a  +  2bt)5t. 

12.    — —  feet  per  second. 

v/3 

CHAPTER  III.    tp.  43) 
J  dy^3(x-l)g(x+l)  c^y^     g-a; 

^^  2x^  ^^    sf2ax-x- 


dx    2s/{x+l){x  +  2) 
(iv.)  ^  =  (x  +  a)^-^(a!  +  /))''-i{(p  +  (/)a;  +  <7a+^6,}. 

Va^  -  a* 

...  3X2  l_a;2 

(XI.)  ..  (Xll.) -. 


(l-a;2)^  (l+a;  +  x2)^(l-a;  +  a;2)^ 

^.  (ii.)     -^0.  (iii.)    :f% 

Va  yo  a'^ 


(i.)  ?^.  (ii.)    _^o.  (iii.)    q3^_!^_0.  (iv.)    -Va. 


ANSWERS  133 

4.    7*96  miles  per  hour.  5.    8  miles  per  hour  ;  4  miles  per  hour. 

8.  ^^^  -yii. 

dv  V 

9.    When  the  pressure  decreases,  the  volume  increases,  and  conversely. 


CHAPTER  IV.     (p.  59) 

1.       (i.)  3sina;cosa;(sinir-eosa;).  (ii. )  sec'*;i;.  (iii.)  ,-• 

cos% 

(iv.)    "1?^^.  (V.)   _2^?-^  .  (vi.)       'l^^-^    . 

sin^x  ( 1  -  sin  x)^  { 1  +  cos  x)"^ 

3.  (i.)  a;*"-^[msin(.T«)  +  «a;"cos(a;")]. 
(ii. )  a;"*~^  [w  cos  (a;")  -  nx"^  sin  (a;")]. 

(iii. )  07"'-^  [m  tan  (ar")  +  ?ix"  sec^  (a;'*)]. 

4,  (i.)  2a;tan-ia;.  (ii.)  sin-^a;.  (iii.) 


2'slx(\  +  x) 

1  _  /J.2  Y 

^^^"^  r+3^T^*  ^^'^  2(i  +  a:2)" 

5.  aw  sin  w<,     aw^  cos  a><. 

6.  a;  =  2awcos2-^;  y=:awsinw^.       x=  -aw^sinw^;  i/  =  au^coab}t. 

The  direction  of  motion  at  time  t  makes  an  angle  ,y  with  the  axis  of  x. 


CHAPTER   V.     (p.  78) 

1.  (i.)  e*(l+a;).  (ii.)'a;"^-ie"^(w  +  ?ia;).        (iii.)  (a  +  hc  +  cax)e''''+<^. 

(iv.)  e^'''""'^|sin-i.T+    ,.J=^V 
V  sfl-xV 

2.  (i.)  2a;ei+^'.  (ii.)  2xe«^'(l+ax2).  (jji)  a;'«-ie«^"(m  +  ?iaa:»). 
(iv.)  a;"*~%*'*(m  +  7ia;"loga). 

3.  (i.)a."^-Ml+mlog.).       (ii.)  (,^(,^2)-       ^"^"^  2;/in- 

(iv)     — r6^_.  (V.)  ^^^!±i±^.  (vi.)  1        . 

^      Nl-a;2)(4-a:2)  '    Xs/^^r:^!  (1-^Wx 

4.  (i.),^     ^^-^  .       (ii.)— ^.  (iii.)    I"^^ 


2v'(2a;+l)(a;-2)'  '    (a2±a;2)f  a:=*(a;-l)** 

/•     \  _vr/i  ■  1        \                 /     \  m?icos(m-»i)a;sin"~^wa7 
(iv.)  ar*(l+log.r).  v.)   ^ '  — 


11.     (i.)  tana.      (ii.)  t&nnd.      (iii.)  -cotyt^.      (iv.)  cot«^.      (v.)  -tan?i^. 

r^-  is  the  tangent  of  the  angle  between  the  radius  vector  to  the 
dr 

point  (r,  ^),  and  the  tangent  to  the  curve  at  that  point. 


134  ANSWERS 

13.     {i.)^l  =  (Sx-l){x-l).     Max.  at  Q,   1^1 

Min.  at  (1,  0). 

(ii. )  -^  =  x{5x-2){x-lf.     Max.  at  origin. 
Mill,  at  (-4,  -03). 

(iii.)  ^  =  2(a;-l)(a;-2)(2a;-3).     Min.  at  (1,  0);  (2.  0). 

Max.  at  (I    ^. 

^'""-^^x^^-^^-      Max.  at  (-1,   -1). 
"^•^  ^        Min.  at  (1,3). 

(V.)  ^  =  2j^P^\.,.     Max.  at  (  -  1,  3). 
dx       (ar  +  x+lr  /      ] 

Min.  at/ 1,^^ 

ax     (x+x  +  i)       Min.  at  (1-4,   --06)  nearly. 

^^"•^  fx=  -  (T\'yHr2r    ^^-  ^'  ^  -  ■^'  "^^^  "'^^^^^• 

ax        (X     i)(x    Z)       Max.  at  (1-4,  18-2)  nearly.      " 

/  ....  dy         2{x'-6x  +  1)       XT    ^-       •  -4. 

(ix.)  ^^^^I^^+^Q.     Max.  at  (I'o,  -1)  nearly. 
dx        (x-4)  Min.  at  (6-45,  9-9)  nearly. 

(x. )  ^^  ^■"^-  ~  ^.     Min.  at  ( 1  -26,  1  -89)  nearly. 
dx       x"^ 

n..  ,       R^  ...  .  R  /•••  \  r.       ^^ ». 

.    (i.    --2-  11.    -•  111.)  Sp=-^dv. 

(iv.)  5y  =  — 5^.  V.)  5«=  -     ^    „ — -dv  +  —  5^. 

p  V     '     ^  ^2  ^ 

19.  —  =  — +  T-+cotC5C. 

A      a      0 


EXAMPLES  ON  THE  PARABOLA,  (p.  83) 


2.    Foci, 

Vertices,  - 

Latera  recta,     - 

Lengths,   -         -         -  1, 

Axes,         -         -         -       x=  -2, 

Tangents  at  vertices,       y=:  -  1, 


(- 

(1) 

(- 

2,    -1), 

y 

3 
4' 

(2) 

(3) 

(-2,  -2), 

(«•  -0 

(-2,  -3), 

(4-^) 

y--2, 

a;-0. 

4, 

1. 

a;=-2, 

1 

^=-2- 

2/= -3, 

1 

ANSWERS  135 


3.    -5.    7.  5.    (1,  2), 


fa2a\ 
\m^    111) 


6.    a;-y-l=0,     a;  +  2/-3  =  0.  7.    x-y  +  a  =  0,     x  +  2/-3a  =  0. 

a;  +  2/  +  a  =  0,     x-  -  y  -  3a  =  0. 

EXAMPLES   ON   THE   ELLIPSE,     (p.  87) 

1.    The  foci,  extremities  of  the  axis,  length  of  latus  rectum,  and  eccen- 
tricity are  for 

(i.)  [±1,0],   [±2,0],    [9±x/3],    3,   \ 

ii.)[2,2],    [0,2],    [3,2],    [-1,2],    [l,2  +  x/3],    [1,2-V3],    3,   ,^- 
(iii.)  [±v^3,  1],   [±2,1],    [0,2],    [0,0],    \   -|- 
(iv.)  [0,  ±1],  [0,  ±2],[±s/3,0],  3,^- 


EXAMPLES  ON  THE   HYPERBOLA,     (p.  90) 

1.  (i.)  (±^^7,0);(±2,0);3;^• 
(ii.)  (1±V7,2);(3,2);(-1,2);3;-^- 

(iii. )  (0,  -  1  ±  v^5) ;  (0,  0) ;  (0,  -  2) ;  8 ;  \/o. 

,-  ,-         4\/3'   \/2l 

(iv.)  (±x/7,0);(±v/3,0);-3-;^- 

2.  (i.)  (±2x/2,  ±2\/2);  (±2,  ±2).         (ii.)  (±2\/2,  t2\/2);  (±2,  ^2). 


CHAPTER  VII.     (p.  104) 

1.      (i.)  ~^^-; ;  3\/a;(3  +  a;);  a;-log(a;  +  3). 

(ii.)  log T>    log^;^ rV'    ^^ — H-^±-7-tan-M — i^-    ' 

^  ^x-\        ^{x-\)  2         ^3  \   x/3   / 


3  ■    2     2    ^^  '    x/3  V   x/3 

(iii. )  sin-i  {2x  -  1 ) ;    2'Jx^-^x  +  2  +  2  log  f  a;  - 1  +  \/a;2  ^"^^  +  2^)  ; 

^{x^  +  x+\)  +  ^\oglx  +  :^  +  slW+xV\\. 

x^  11 

2.    x^\n-'^x  +  sll-x^;     -^tan~^a;-pa;2  +  ^log(l  +  a:2) ; 
o  o         b 

,     /l-8x2\       .     ,      X 
cos  4a;  (  — ^2 —  )  +  sni  4a;  •  ^  ; 

/9x2-2\    .    .,       2a;        ^         a;"*+i  a;"*^!  ^,  ,     , 


136  ANSWERS 

,     1,      /  (l+a;)2  \       1  ,2a;-l  .         Ill,      fx+\\ 

8.    log  tan  2  •     log  tan  -  ^-  •     -^  log  tan  (^  ^  + 


1  /-s s 7^,        1,      2secx--l 

-  log  (a  tan  ^  +  Va^  tan^^  +  ¥).     ^  log  2  ^^^i ' 

11.     /  ^- — i =  0  tan-i        tan  j.    .  /  ^ — ^ =  k  tan-^  I  3  tan  - 

j5  +  4cosa:    3  \3        2/  ./5-4cosx    3  \  2/ 

■      f       dx       ^Ij      ^  +  ^^"1  /•       tZx     ^1^^    ^  +  ^^^"1 

J4  +  5cosa;     3°^3_^^^|'  j4-5cosx    3"gj_3^^^^| 


,-  /a;     TT 

;-  _dx       ^2j     ^g  +  tonj^g-j^ 

J2  +  3Bi„:.    '^5"'''^/5-tan(|-^) 


1  -  \/5  tan 


r    dx    ^  2  ^^  ^"""-""(I'i) 

J2-3sina;     ^  "^"~T      /x     ttV 
l+VotanI  2~4  ) 


,«  1       o.    .    .     2   .    ,      cos^^sin^     3        ^   •    ^     3^ 

12.  ^  cos^^  sin  0  +  ^  sin  ^.     j +  -  cos  ^  sin  ^  +  ^  ^. 

...  a;"    .  n        ,  w(w- 1)  r  «  <>  7 

15.  —  sin  mx  +  — s  x"-^  cos  wa; 5—  /  a;"--^  cos  wa:  dx. 


CHAPTER  VIII.     (p.  126) 

1.  (i.)  1.  (il)^fe-^+l\  (iii.)  aMog^^. 
(iv.)  64.                          (V.)  3(^2^-1). 

2.  -  :  the  difference  between  the  area  bounded  by  the  a:-axis,  the  y-axis 
6 

and  the  curve,  and  the  area  which  lies  on  the  negative  side  of  the 
a;-axi8. 

«    343 
^'    12' 


ANSWERS  137 

*•     <^-)     -3  -•  ^"-^  ^'   12-'   IrT- 

(iii.)  !:^',   '^«',   '[?^.  (iv.)  aMog^^^. 

^      '     8  '    12'    471  ^      ^  ^tan^i 

/     \  *''f    -1  *^(t'*ii^2~  tan^i)       a6 1      (&  +  «  tan  d^) {b-a  tan  g^) 
IV.)  —  an     -^2tan^jtan^2+l>2'     T  ^^{b  +  atai\d^b-ata.ne^)' 

6.  (i.)  a8eca(e-'''°^"-l).  (»•)  |((27rVT+472)  +  iog(2,r  +  v/r+47r2)). 
(iii.)|(e-e-i). 

7.  (i.)  g.  (11.)  487r.  (ill.)  — -. 

(iv.)  -7r/iV.  (v.)  2a62,r2.  (vi.)  -irabc. 

5  3 

11.     (i. )  ~l  from  that  end. 
o 

(ii.)  On  the  radius  bisecting  the  arc  at  a  distance  from  the 


centre. 

'""37r' 


,...  ,  -     46       _     4a 
(ill.)  x=     ,     y 


(iv.)  On  the  radius  bisecting  the  sector  at  a  distance  _  ?HL^  from 
the  centre. 

(v.)  On  the  bisector  of  the  chord  at  a  distance  -a ^^^  "• from 

^,  ^  3     a  -  sin  a  cos  a 

the  centre. 

(vi. )  The  middle  point  of  the  radius  perpendicular  to  the  base. 

12.     (i.)  lM^2.(rodof  length2/).         (ii.)  iMa^.  (iii.)  iMa^. 

3  JL  Jd 

(iv.)  f  (a'^  +  in-       (v.)^Ma=.        (vi. )  (a)  '^'5' :    (,3)  M (?^-^| 
(vii.)  (a)  M^:    (m  M^:    (y)  M^'^). 
(viii.,  M(-±£?).  (ix.,  M(-|.-). 


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